Odysseus, A Hero Essay - 1135 Words

Is there such a thing as a true hero? Or are those that are considered heroes just regular people who made the right choice at the right time and became idolized for it? To be a true hero, the person would have to be totally good. It is impossible for a human being to be totally good because weaknesses, character faults, and the tendency to make mistakes are all rooted deeply into human nature. Therefore, no human being can ever truly be a hero, though we may do heroic deeds. A well known example of such a person is Odysseus from Homers Odyssey. Odysseus is idolized for his few heroic deeds during the Trojan War and his journey home to Ithaca. He is often thought of as a hero, but, as he is human and therefore subject to human†¦show more content†¦Even in the bleakest of situations, Odysseus did not give up. For example, when the crew landed on the Island of the Sun, Odysseus, who knows that his crew is prophesied to perish there, warns his shipmates that their stores of foo d are on their ship and that they musnt eat the catle of the sun god, Helios, or they will all pay dearly for it. To the very end, Odysseus was dedicated to his companions and even tried to save their lives, although Odysseus already knew they were prophesied to perish. It was this dedication to quest and companions that made Odysseus and ideal leader. Together, these and Odysseuss other positive traits and deeds may lead people to consider him a hero. Now, if Odysseuss character could be defined by only his admirable qualities and heroic deeds, then he could be considered a true hero. Alas, there is so much more to Odysseus than just that. He also possesses some other, less than favorable characteristics. Odysseus does not fit the mold of a true hero because he is not perfectly of good character and also, like all human beings, possesses some less than favorable qualities. What I mean by less than favorable is that the qualities are not neccessarily those that you would associate w ith the perfectly good being. For example, you probably do not associate the characteristics merciless, reckless, or cruel with your favorite fictional hero.Those are characteristics of the classic villain, arent they? My point is, each of thoseShow MoreRelatedThe Hero Of Odysseus900 Words   |  4 PagesRogers Some people think of a hero as someone with courage, determination, or someone that have risked their lives for others. Others think of a hero as someone who has outstanding strength and is clever. A hero, not only has physical strength, but also has mental strength as well. Odysseus, ruler of Ithaca, proves he is a hero by having all of these traits. Odysseus fights through hard times and overcomes obstacles, he kept his men together for most of the story, and he avenges himself whenRead MoreOdysseus Is A Hero?950 Words   |  4 Pagesmost people, Odysseus is believed as a hero. A hero is a person who is admired or idealized for courage, achievements, and noble qualities. In one of Homer’s classic epics, The Odyssey, Odysseus is admired by many people as a man who has intelligence, nobility, and confidence. However, women who both loved and knew Odysseus than anyone, thought of him differently and knew him in depth. Through The Meadowlands and The Penelopiad, Circe and Pene lope showed their strong opinions of this â€Å"hero† through theirRead MoreOdysseus Is A Hero?1143 Words   |  5 PagesHomer, The character Odysseus is one of the greek heros in this book. He is known as a great hero, because he manages to get through all of these dangerous mission such as : First odysseus makes it alive after travelling for ten years and facing different and more difficult challenges. Odyssey faces dangerous creatures and people. Homer have odysseus pass all theses task and missions to name him one of the great greek gods that ever lived . Some may say Odysseus is not a hero but why would HomerRead MoreOdysseus A Hero730 Words   |  3 PagesIs Odysseus a Hero? Heroes are often thought of as great figures that conquer evil, kill the monster, and save the day. Odysseus is often referred to as a hero. He is a strong individual striving to complete his goal, return to his wife and son and remove suitors that have taken his home. Although he is seen as a hero by definition and he appears to be one as well there are decisions he made that may not be truly heroic. In The Odyssey by Homer, Odysseus decisions to value his crews life, andRead MoreIs Odysseus Not Hero?1706 Words   |  7 Pageshearts. Many people argue that Odysseus is not hero but considering the things he does through his journey his characteristics began to show a little of his heroic side. In The Wanderings of Odysseus from the Odyssey a epic poem by Homer, Odysseus is on his way back to Ithaca his home island after winning the ten year old Trojan War to come see his wife, Penelope and his son, Telemachus. The journey to home takes a very unexpected turn for Odysseus and his m en. Odysseus is challenged with many obstaclesRead MoreOdysseus Is Not A Hero1965 Words   |  8 PagesMerriam-Webster defines a hero as, â€Å"A person who is admired for great or brave acts or fine qualities.† For Odysseus to be a hero this definition is supposed to fit and describe him and his character, but why does this definition not describe or fit him and his character? This is because Odysseus is not a hero. Odysseus has done unethical things on his journey that makes him a person that should go unadmired. For example, when Odysseus ignored the death of his man in order to make more progress onRead MoreLoachus : A Hero : Odysseus As A Hero737 Words   |  3 PagesOdysseus is a Hero There are countless ways to describe a hero. A hero is someone who fights for what they believe in. They fight with their men and are determined to come back home. In the Odyssey, Odysseus is a hero because he is a leader, very determined, and is extremely skillful. Odysseus was a leader in various ways. He led his men through the Trojan War and many difficult challenges that faced them. Sometimes he didnt want so many men walking in on something unexpected, so he only took aRead More Odysseus: A Hero Essay1333 Words   |  6 PagesOdysseus: A Hero Heroism was not an invention of the Greeks. Yet, through the first hundreds years of their civilization, the Greek literature has already given birth to highly polished and complex long epics that revolved around heroes. These literature works gave many possibilities of definition of heroism. The Greeks illustrated heroism to obey the rules laid down by the gods and goddesses, and those who obey the rules would gain honor and fame. The Greeks regarded intelligence as oneRead MoreOdysseus: the Anti-Hero1534 Words   |  7 PagesOdysseus: The Anti-Hero Throughout Homer’s epic The Odyssey, Odysseus is a hero. In all myths and legends, a hero combats the â€Å"monsters.† In the typical story a hero is unselfish and fights to protect his people while the monsters are greedy powerful things that antagonize the people. Yet despite this typical storyline, if we read closely, we may conclude that Odysseus is actually the oppressive hero in many of these situations – provoking the â€Å"monsters† into fighting. The â€Å"monsters† in the OdysseyRead MoreOdysseus As A Great Hero869 Words   |  4 PagesTen years after the fall of Troy, Odysseus a great hero has yet to return to his home in Ithaca. It begins with Athena and Poseidon who helped the Greeks during the Trojan War. Athena turned against the Greeks and convinces Poseidon to do the same. The Greeks are hit by storms on the way home and many ships are destroyed and the fleet is scattered. The war and his distress at sea keep Odysseus away from Ithaca for twenty years. While Odyesseus was gone his son Telemachus has grown into a man and

The Effects Of Microaggressions On An Individual - 1167 Words

Psychiatrist Chester Pierce, MD in the 1970’s was the first to create the term microaggressions (Sue, 2010). While Dr. Pierce was the first to coin the term, he was not technically the first to start it. In fact, the idea of microaggressions was also introduced in the specific work of Jack Dovidio, PHD (Yale University) and Samuel Gaertner PHD (University of Delaware) when formulating aversive racism (Sue, 2010). Aversive Racism is defined as the following: â€Å"Many well-intentioned Whites consciously believe in and profess equality, but unconsciously act in a racist manner, particularly in ambiguous situation† (Sue, 2010). Similarly, microaggressions is defined as: â€Å"Microaggressions are the everyday verbal, nonverbal, and environmental slights, snubs, or insults, whether intentional or unintentional, that communicate hostile, derogatory, or negative messages to target persons based solely upon their marginalized group membership† (Wiley Sons, 2010). The harmful effects of microaggressions on an individual are derived from how subtle and indirect the statements are. Derald Sue, clarified that microaggressions are damaging to the person who is experiencing them because while they feel insulted they are not sure if the perpetrators are aware (Sue, 2010). Thus, people having these experiences are caught in a Catch-22 (Sue, 2010). The subtly of microagressions is what puts people in a â€Å"psychological bind† (Sue, 2010). Sue believed that in order to understand the severity ofShow MoreRelatedThe Effects Of Microaggressions On An Individual854 Words   |  4 PagesPsychiatrist Chester Pierce, MD in the 1970’s first introduced the term microaggressions (Sue, 2010). While Dr. Pierce was the first to coin the term, he was not technically the first to start it. In fact, the idea of microaggressions was also introduced in the specific work of Jack Dovidio, PHD (Yale University) and Samuel Gaertner PHD (University of Dela ware) when formulating aversive racism (Sue, 2010). Aversive Racism is defined as the following: â€Å"Many well-intentioned Whites consciously believeRead MoreFactors Affecting A Learning Community820 Words   |  4 Pagesperformed (Saunders, 2008). Thus, using these findings from the article, one can conclude that microaggressions affects a persons well being to a degree that can affect them perform certain tasks. As the researcher found, students not part of a learning environment faced the abuse of microaggressions, which ultimately turns the individual away from learning in general. Thus, microaggressions will effect integral parts of ones life and influence them to negatively perform on major and minor tasks.Read MoreAnti Heterosexuality And Its Effects On Society1369 Words   |  6 PagesInvisible heterosexism is the cause of all these atrocities because it fears disruption from the heteronormative system and leads to detrimental effects for those who are target of its abu se (190). Some of the microaggressions they receive are when those from this group are oversexualized and rid of other aspects of their humanity (192). More blatant microaggressions stem from an actual fear, rather than loathing, of a mythical non-heterosexuality contagiousness, or the idea that, this with mere interactionRead MoreThe Endangered And Endangered Species : Diversity1685 Words   |  7 PagesNewspapers and magazines have published articles that highlight the different benefits of maintaining a diverse group of individuals in schools, communities and organizations. Nevertheless, despite this increasing recognition and celebration of the countless advantages that racial diversity has brought to our day-to-day life, the emotional and psychological wellbeing for individuals of different racial groups is very often being overlooked by our society. In the ongoing discussion of the issue of racialRead MoreCritical Race Theory And Social Darwinism1248 Words   |  5 Pagesminorities must have deserved their situations because they were â€Å"less fit† than those who were better off. †¢ WHITENESS Whiteness is a complex and fragmented identity, which involves privileges for white individuals and groups and discrimination, marginalization and oppression of non-white individuals and groups. Whiteness is a racial privilege from which all white Australians benefit and receive unearned social benefits as part of legacy of racial system of wealth and privilege. †¢ WHITE PRIVILEGERead MoreThe Implication Of Racial Microaggressions1772 Words   |  8 Pagesimplication of racial microaggressions in daily life. Three journal articles and a book chapter are explored in an effort to obtain a greater understanding of the effects of racial microaggressions experienced by people of color and to bring light to how often racial microaggressions are committed by White Americans without notice or accompanied by attempts to explain away the offenses. The sources used provide examples of obvious acts of discrimination as well as subtle microaggressions which are oftenRead MoreAt College And Universities Across America, Students Are1204 Words   |  5 PagesAt college and universities across America, students are being constrained to an increased sense of political correctness. This is because students are demanding protection from microaggressions. Microaggressions are words and ideas that seem to have no malicious intent from the outside, but ar e viewed as a kind of violence nonetheless. Furthermore, professors now need to be concerned about trigger warnings and avoid course content if they believe it may cause a strong emotional response. For instanceRead MoreOffense Taken: Microaggressions in Society Essay1902 Words   |  8 Pagesof counseling psychology at Colombia University. He has solidified the definition of microaggressions as â€Å"†¦brief and commonplace daily verbal, behavioral, or environmental indignities, whether intentional or unintentional, that communicate hostile, derogatory, or negative racial slights and insults toward people of color† (Sue 271). Due to Sue’s work toward refining and reintroducing the term, â€Å"microaggressions† has spread to college campuses and intellectuals that have validated and have even appliedRead MoreHealth Care Case Study776 Words   |  4 PagesIn 2015, our hospital spent $102,037,333 on charity care and uncompensated care. We support those who are unfunded or low-income. Since we help every single individual that walks through our doors, we have no biases. It is a random selection based off the community and surrounding communities. Whether you are homeless, rich, white, black or anywhere between those, our services are available to you. We have a large support system and services for the elderly. A possible barrier is language. Most ofRead MoreI Am A Woman With Amniotic Band Syndrome1348 Words   |  6 PagesThese microaggressions are not meant to be hurtful, but they are insulting occurrences that do happen. I receive daily macroaggressions towards my race more than I do with my disability. Many people like to point out that when I speak Spanish, how I sound very â€Å"Latina† and how different it is compared to my â€Å"white valley girl† accent that I have when I am speaking English. There are many studies that have research on microaggressions towards one singular identity instead of microaggressio ns towards

Yugoslavia Essay Research Paper One of the free essay sample

Yugoslavia Essay, Research Paper One of the youngest states of Europe, Yugoslavia was created after World War I as a fatherland for several different rival cultural groups. The state was put together largely from leftovers of the collapsed Ottoman Empire and Austria-Hungary. Demands for self-government by Slovenes, Croats, Serbs, and others were ignored. Yugoslavia therefore became an uneasy association of peoples conditioned by centuries of cultural and spiritual hates. World War II aggravated these competitions, but Communist absolutism after the war controlled them for 45 old ages. When the Communist system failed, the old competitions reasserted themselves ; and in the early 1990s the state was rent by secessionist motions and civil war. Within several old ages these struggles had drastically altered the size of the state. As it existed in 1990, Yugoslavia was bounded on the North by Austria and Hungary, on the nor-east by Romania, on the E by Bulgaria, on the South by Greece, and on the West by Albania, the Adriatic Sea, and Italy. We will write a custom essay sample on Yugoslavia Essay Research Paper One of the or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page It was 600 stat mis from north to south and 250 stat mis from West to east at its widest portion. Its entire country was 98,766 square stat mis. Three old ages subsequently the state s country had been reduced by 60 per centum and its population of 23 million cut by more than half. The states of Slovenia, Croatia, Macedonia, and Bosnia and Herzegovina had seceded, go forthing Serbia and Montenegro as the Federal Republic of Yugoslavia. The description below screens Yugoslavia, as it existed prior to decomposition. Yugoslavia has a cragged terrain. The northwesterly country consists of the Karawanken and Julian alps in Slovenia. The latter scope contains Mount Triglav at 9,396 pess. The Dinaric Alps occupy much of the West with extremums making more than 8,000 pess. To the south the Sar Mountains and next scopes belong to the Rhodope massif, which extends due south into Greece. The major country of flatland prevarications in the nor-east and is portion if the big Mid-Danube, or Pannonian, Plain. Along the shore of the Adriatic Sea is a little coastal field known as the Dalmatian seashore. The longest river in Yugoslovia is the Sava, which flows from the Austrian boundary line due east for 584 stat mis to fall in the Danube at Belgrade. The Danube flows for 367 stat mis through Croatia and Serbia. Its major feeders are the Sava, Drava, Tisa, and Morava. Other rivers are the Drina, Bosna, Neretva, and the Vardar. There are more than 200 lakes of which the largest is Lake Scutari on the Albanian boundary line. The cragged nature of the state causes considerable climatic differences from one topographic point to another. The Dalmatian seashore has a typical Mediterranean clime with hot, dry summers and mild, showery winters. The Mid-Danube Plain has a Continental clime with cold winters, hot summers, and moderate precipitation. The mountain parts have on the whole colder and shorter summers and more terrible winters compared with other parts. The major environment jeopardy in Yugoslovia is temblors. The whole part is capable to temblors of considerable badness, and in 1963 the metropolis of Skopje was about wholly destroyed by one. Yugoslavia has legion sedimentations of brown coal, or brown coal, exist, but there is small good-grade black coal. There are some little crude oil and natural gas field. The major beginning of energy is waterpower, which provides about one tierce of the state s electricity. Yugoslavia is a major European manufacturer of lead and Cu. Other minerals include Fe ore, Zn, Ag, gold, nickel, quicksilver, and Sb. About 37 per centum of the state is forest covered. The prevailing species are oak, beech, and other deciduous trees, with such evergreens as pine and fir in the mountains. The dirts of the Mid-Danube Plain are the best in the state for farming. Yugoslavia has a broad scope of animate beings, including cervid, foxes, wolves, Canis aureuss, bears, and seldom, lynxes. Birds include grouse, partridge, swans, turkey vultures, peckerwoods, and pelicans. The Adriatic Sea contains anchovies, pilchards, mackerel, tuna, and other fishes. The dwellers of Yugoslavia were of varied cultural beginnings. Harmonizing to the 1981 nose count the largest group was the Serbs, who numbered 8.1 million, or 36 per centum of the population. Like the bulk of Yugoslavs, they speak a Slavic linguistic communication. They belong to the Eastern Orthodox Church. The Croats totaled 4.4 million, 20 per centum of the population. They speak a linguistic communication similar to that of the Serbs but are Roman Catholics. The Slovenes totaled 1.8 million and lived in the northwesterly corner of the state. They have their ain Slavic linguistic communication and are Roman Catholics. Other Slavs include the Macedonians 1.3 million and the Montenegrins 600 1000. Both groups are Eastern Orthodox. There were about 2 million Muslims, descended from Slavs who converted to Islam during the long Turkish business. In 1991 the nose count showed a entire population of 23,475,887, with the cultural proportions staying about the same. A non-Slavic people, the Albanians, live in the southern portion of the state. They figure about 1.7 million and are the fastest-growing cultural group. Many are Muslims. There are smaller groups of Hungarians, Romanians, Bulgarians, Slovaks, Czechs, and others who live largely in the northeasterly state of Vojvodina. The largest metropolis is Belgrade, in Serbia, with about 1.6 million dwellers. Zagreb, in Croatia, is the 2nd largest, with over 930,000. Other metropoliss with more than 250,000 dwellers are Skopje, Sarajevo, Ljublijana, and Novi Sad. Yugoslav civilization has been influenced by the Slavs, Turks, Italians, and Austrians. In general the impact of the long Turkish business is seen in the nutrient, common people costumes, and architecture of many of the people. Modern Yugoslav art is best known for it sculpture. Ivan Metrovic achieved universe celebrity for his dramatic statues. He spent the ulterior old ages of his life in the United States and had a figure of talented followings, including Anton Augustin. Such contemporary painters as Mila Milunovic, Petar Dobrovic, and Milan Konjovic have been influenced by the Gallic school. The earliest Yugoslav literature was spiritual in nature. The first popular literature appeared in medieval Serbia, chiefly in the signifier of heroic poem verse forms depicting the battle of the Serbs against the Turks. These verse forms were chanted by folk singers who traveled from small town to village. In Dubrovnik and other topographic points on the Dalmatian seashore, a more sophisticated literature influenced by the Italian Renaissance developed. Poetry and play were peculiarly popular. In the nineteenth century Serbian authors laid the foundations of a modern literature. Vuk Karadzic reformed the linguistic communication and collected common people poesy, while Petar Petrovic Njego produced heroic poesy on the subject of autonomy. In Croatia, Ljudevit Gaj, and in Slovenia, France Preeren, were taking figures in the development of their peoples literature. Among modern authors are the Serbs Branimir Cosic, Branko Opic, and Ivo Andric, whose novel The Bridge in the Drina has been translated into many linguistic communications. The Croat authors Vladimir Nazor, Miroslav Krleza, and Slavko Kolar are besides popular. Yugoslav folklore is really colourful. Each cultural group has its ain costumes, vocals, and dances. The most popular common people dance is the kolo, a circle dance performed to lively music. Soccer is the most popular athletics, and Yugoslavia has produced some star participants. Yugoslav hoops squads have besides had some success in international competitions. Winter athleticss are popular, particularly in the Alps of Slovenia. The winter Olympic games of 1984 were held in Sarajevo in Bosnia and Herzegovina. Farming is a major business, using about 29 per centum of the labour force. Most farms are owned in private and are little. The major harvests are maize, wheat, barley, oats, murphies, sugar Beta vulgariss, helianthuss, baccy, and alfalfa. About one tierce of the agricultural country consists of grazing lands for croping sheep, hogs, and cowss. Much of this is in the mountains. The turning of fruit includes plums, apples, Prunus persicas, pears, apricots, Cydonia oblongas, and cherries. Figs and olives are grown chiefly along the seashore. Grapes are widely grown and vino produced, some for export. Fishing is carried out along the Adriatic seashore and on the Danube River. Forestry is concentrated manfully in the mountain woods of the northwest. It supports mush and paper and furniture industries. Much of Yugoslavia s industry is located in the Northwest, where it was originally established by the Austrians. The oldest Fe and steel works is at Jesenice in Slovenia, and the largest is at Zenica in Bosnia and Herzegovina. The processing and refinement of metallic minerals gives considerable employment. There is an technology industry based chiefly in the North and around Belgrade. The car industry has been developed with foreign assistance. A little, low-priced auto called the Yugo, based on an Italian Fiat theoretical account, is manufactured for export. Ships are built in the Adriatic ports of Rijeka, Pula, and Split. Fabrics and chemicals are besides produced, and fruits, fish, and baccy are processed. Factories and other economic endeavors in Yugoslavia have non been run by the province as in other Communist states. They are operated by workers councils, which compete with one another for clients and advertise as in the West. The jobs of rail building in a cragged state such as Yugoslavia have favored the development of a main road web. There are two major main roads one running from the Austrian boundary line to Greece and the other along the Adriatic seashore. The latter is used by the big Numberss of tourers who visit the coastal metropoliss and resorts. Tourism is a major beginning of foreign income. The major ports are Rijeka, Split, PloCe, Koper, and Bar. Yugoslav Airlines is province owned and flies to many foreign finishs. The chief international airdromes are at Belgrade and Zagreb. Postal, telegraph, and telephone services are run by the province. Radio and telecasting broadcast medium are besides under the control of a province organisation. Education is mandatory between the ages of 7 and 15. All instruction is free, including that at the university degree. There are particular schools for the smaller cultural minorities. Each democracy has its ain university. The Socialist Federal Republic of Yugoslavia consisted of six democracies: Bosnia and Herzegovina, Froatia, Macedonia, Montenegro, Serbia, and Slovenia. The democracy of Serbia contains the independent states of Kosovo and Vojvodina. Each democracy and state had its ain fundamental law and assembly. Local personal businesss were handled by smaller assemblies. At the top of this system of assemblies was the Assembly of the Socialist Federal Republic of Yugoslavia, which was divided into two Chamberss the Federal Chamber and the Chamber of Republics and Provinces. There was a State Presidency with nine members. It functioned as a corporate presidential term with a president at its caput. The place of president rotated every twelvemonth among the representatives of the democracies and states. Until 1990, merely one political party, the League of Communists, was permitted. Any effort to organize parties based on cultural beginnings was strongly opposed. The ascendants of the Yugoslavs appeared in the part in the seventh century. The Slovenes formed a little province that was absorbed by the ninth century by the Franks, a Germanic people. The Croats developed an independent province under King Tomislav at the beginning of the tenth century. At the terminal of the 11th, nevertheless, Croatia came under Magyar control. By the twelfth century the Serbs had established a powerful province, and the fourteenth century Stefan Duan, male monarch of Serbia, extended his imperium to include Macedonia and much of Greece. A major catastrophe overtook the South Slavs with the Turkish invasion of southeasterly Europe in the 15-century. Turkish control of the part lasted for five centuries. At the same clip Slovenia and Croatia became portion of the Austrian Hapsburg Empire. Rebellions broke out at assorted times. In 1555 the Slovenes and in 1573 the Croats revolted against Hapsburg regulation to no help. In 1804 the Serbs rose against the Turks under their national hero, Karageorge, and once more in 1815 under Milo Obrenovic. In 1830 Serbia won partial independency from Turkey with Obrenovic as male monarch, and in 1867 full independency was achieved. During the period of Turkish control, the little province of Montenegro maintained its independency. The city state of Ragusa ( now Dubrovnik ) besides remained free of foreign control by adept diplomatic negotiations. In 1812 and 1913 Serbia was winning in the Balkan Wars against Turkey and Bulgaria. In 1914 the blackwash of Archduke Francis Ferdinand of Austria by a Serb gave Austria-Hungary the alibi to declare war on Serbia, and event that led to World War I. After the war the dissolution of Austria-Hungary made possible the creative activity of a new province for the South Slavs. In 1918 the Kingdom of Serbs, Croats, and Slovenes was proclaimed ; it was renamed the Kingdom of Yugoslavia in 1929. The land endured as an uneasy alliance of reciprocally hostile cultural groups. In 1939 an understanding was reached to give Croatia liberty, but in 1941 Yugoslavia was invaded by Germany, Italy, and Hungary. Serbs resisted the business forces, and the Communist Partisans under the leading if Josip Broz, known as Tito, became the dominant group. The Croats and Slovenes, nevertheless, sided openly with Germany and Italy. In 1945 the state became a democracy with the Communists as its swayers. Although Serbs remained the dominant population, Tito himself was half Croat and half Slovene. In 1948 Yugoslavia was expelled from the Soviet axis for declining to subject to Soviet orders. Tito managed successfully to maneuver a nonaligned way between the two world powers, the Soviet Union and the United States. After Stalin died in 1953, this undertaking became easier. Then Tito died in 1980, and the delicate federation he had held together began to unknot. The League of Communists relinquished their constitutionally guaranteed monopoly on power, and in 1990 the first free multiparty elections were held since Tito took power. In May 1991 Serbia and its Alliess blocked the election of a Croat to the federal presidential term, go forthing the state without a president. A new Serbian leader emerged Slobodan Milosevic, who renewed the antique promise of a Greater Serbia. This end entailed taking parts of other democracies where Serbian minorities lived and unifying them with Serbia. On June 25, 1991, Croatia and Serbia declared their independency from Yugoslavia. Federal military personnels made up largely of Serbs poured into Slovenia, resisted by Slovenian reserves. The Serbs invaded Croatia. At the terminal of 1991 Germany, followed by the European Community and the United States, recognized the independency of Croatia and Slovenia. A cease-fire went into consequence, go forthing Slovenia and Croatia mostly at peace fro the clip being. But Serbia had taken approximately one tierce of Croatia s district. The force spread following to Bosnia and Herzegovina. Early 1992 the democracy voted for independency, but the big Serb minority boycotted the referendum. Recognition by the European Community and the United States followed in April. A new Yugoslavia, made up of Serbia and Montenegro, was proclaimed in April of 1992. Meanwhile, a civil war had erupted throughout the democracy as Serb reserves shelled metropoliss and towns. The state of affairs in Bosnia was complicated by spiritual differences. Many of its occupants, Serb and Croat likewise, were Muslims. Serbs tended largely to be Serbian Orthodox, while Croatians were largely Roman Catholic. These competitions added to the cultural hates. Croat and Serb Christians besides turned their arms on the Muslim minority. A run of terrorist act and race murder, which they termed cultural cleaning, was started by the Serbs against Muslim. Many Muslims were killed outright. Muslim adult females were raped, and work forces and male childs were put into concentration cantonments. At least two million people became refugees, and about 140,000 were losing presumed dead. By the terminal of 1992, Serb forces had occupied more than 70 per centum of Bosnia. Many of its metropoliss were in ruins, among them Sarajevo, the capital. The United Nations imposed economic countenances but obtained no peace colony. Croatia and Serbia had determined to split Bosina between them, go forthing little enclaves for Muslims to populate. In Serbia itself the countenances had created mayhem. Hyperinflation was running at the unparalled per centum rare of quadrillions per twelvemonth, presenting a menace to the endurance of the province. 31c

Hamlet Is He Insane Essay Example For Students

Hamlet: Is He Insane? Essay The term insanity means a mental disorder, whether it is temporary or permanent, that isused to describe a person when they dont know the difference between right or wrong. Theydont consider the nature of their actions due to the mental defect.(Insanity, sturtevant) InWilliam Shakespeares play Hamlet Shakespeare leads you to believe that the main character,Hamlet, might be insane. There are many clues to suggest Hamlet is insane but infact he iscompletely sane. Throughout the play Hamlet makes wise decisions to prove he is not insane. He knowsexactly what he is leading up to. He just delays to act due to his indecisiveness. An example ofthis is in Act III, section III, line 73, Hamlet says Now might I do it pat, now a is a-praying, andnow Ill do it-and so a goes to heaven, and so am I revenged that would be scanned. A villainkills my father, and for that, I, his sole son, do this same villain send to heaven. Why, this is hireand salary, not revenge. He says here that he has his chance to kill his fathers murder but, he ispraying. By killing him while hes praying his soul goes to heaven and this wouldnt be revenge. This is not a thought of an insane person. An insane person would have completed the murder atthis opportunity. In Act III, scene I, line 55, To be or not to be, Hamlet displays hisindecisiveness by thinking about suicide because of the situation he is in. He would rather bedead than live with the thought of his fathers death goin g unavenged. He is scared to getrevenge because he found out from a ghost and he doesnt know what to do. In line 83, Thusconscience takes a major part in the thought and action of murder. This is why he delays so longto commit the murder. An insane person would not wait. They would be more apt to act inimpulse. Hamlets madness only existed when he was in the presence of certain characters. WhenHamlet is around Polonius, Claudius, Gertrude, Ophelia, Rosencrantz, and Gildenstern, hebehaves irrationally. For example in Act II, section II, Polonius asks Hamlet, Do you know me,my Lord? Hamlet replies, Excellent well, you are a fischmonger. Hamlet pretends not toknow who Polonius is, even though he is Ophelias father. When Hamlet is around Horatio,Bernardo, Fransisco, the players and the Gravediggers, he behaves rationally.In Act I, sectionV, lines 165-180, Hamlet says How strange or odd someer shall think meet to put on anticdisposition), That you, at such times, seeing me, never shall, with arms encumbered thus, or thishead shake, . He dowsnt want Horatio to reveal anything that might be going on. If Horatioisnt surprised by Hamlets supposed madness or he leads on that he knows or something thenHamlets antic disposition will not be affective. He tells Horatio he will be acting mad and hemustnt say anything. If Hamlet plans to put on an act of antic disposition then he cant beinsane. Throughout the play Hamlet questions everyone. He questions the ghost, is he real? Hequestions Rosencrantz and Gilderstern, Polonius, Claudius, Gertrude, and Ophelia. In Act III sc. I line 103, Hamlet asks Ophelia Ha, Ha! are you honest? Are you fair? and Where is yourfather? Ophelia tells Hamlet hes at home. Hamlet somehow knows that he is being spied onby Claudius and Polonius so he pretends to be mad. At this point the King says in line 180Madness in great ones must not unwatched go. He still is not completely convinced ofHamlets madness so he is cautious. .u8d95fd691c377e603c8036a5fcc9c4fa , .u8d95fd691c377e603c8036a5fcc9c4fa .postImageUrl , .u8d95fd691c377e603c8036a5fcc9c4fa .centered-text-area { min-height: 80px; position: relative; } .u8d95fd691c377e603c8036a5fcc9c4fa , .u8d95fd691c377e603c8036a5fcc9c4fa:hover , .u8d95fd691c377e603c8036a5fcc9c4fa:visited , .u8d95fd691c377e603c8036a5fcc9c4fa:active { border:0!important; } .u8d95fd691c377e603c8036a5fcc9c4fa .clearfix:after { content: ""; display: table; clear: both; } .u8d95fd691c377e603c8036a5fcc9c4fa { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u8d95fd691c377e603c8036a5fcc9c4fa:active , .u8d95fd691c377e603c8036a5fcc9c4fa:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u8d95fd691c377e603c8036a5fcc9c4fa .centered-text-area { width: 100%; position: relative ; } .u8d95fd691c377e603c8036a5fcc9c4fa .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u8d95fd691c377e603c8036a5fcc9c4fa .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u8d95fd691c377e603c8036a5fcc9c4fa .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u8d95fd691c377e603c8036a5fcc9c4fa:hover .ctaButton { background-color: #34495E!important; } .u8d95fd691c377e603c8036a5fcc9c4fa .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u8d95fd691c377e603c8036a5fcc9c4fa .u8d95fd691c377e603c8036a5fcc9c4fa-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u8d95fd691c377e603c8036a5fcc9c4fa:after { content: ""; display: block; clear: both; } READ: Biography: During his few weeks as Vice President, EssayThroughout the entire play Hamlet is careful with his actions. He thinks everythingthrough. Although he delayed his actions longer than Laertes did, he planned all his actions outinstead of acting out in a foot of rage. He had to be completely sure before taking action. Hamlet was completely aware of his actions and what was morally correct. He never lost sightof his objective to expose the Kings sin of murdering his father and obtaining revenge. Hamletwas completely sane throughout this Shakespearean tragedy. Category: Shakespeare

Asynchronous Transfer Mode Essay Research Paper Asynchronous free essay sample

Asynchronous Transfer Mode Essay, Research Paper Asynchronous Transfer Mode Asynchronous Transfer Mode: Asynchronous Transfer Mode By Gene Bandy State Technical Institute Asynchronous Transfer Mode: Asynchronous Transportation Mode ( ATM ) is a # 8220 ; high-velocity transmittal protocol in which information blocks are broken into little cells that are transmitted separately and perchance via different paths in a mode similar to packet-switching engineering # 8221 ; . In other words, it is a signifier of informations transmittal that allows voice, picture and informations to be sent along the same web. In the yesteryear, voice, picture and informations were transferred utilizing separate webs: voice traffic over the phone, picture over overseas telegram webs and informations over an internetwork. ATM is a cell- shift and multiplexing engineering designed to be a fast, general intent transportation manner for multiple services. It is asynchronous because cells are non transferred sporadically. Cells are given clip slots on demand. What seperates ATMs is its capableness to back up multimedia and incorporate these services along with informations over a signal type of transmittal method. We will write a custom essay sample on Asynchronous Transfer Mode Essay Research Paper Asynchronous or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page The ATM cell is the informations unit used to convey the information. The information is broken into 48-byte informations packages for transmittal. Five bytes of control informations are appended to the 48-byte informations packages, organizing a 53-byte transmittal frame. These frames are so transmitted to the receiver, where the 5-byte control informations ( or Heading ) is removed and the message is put back together for usage by the system In an ATM web, all informations is switched and multiplexed in these cells. Each ATM cell sent into the web contains turn toing information that achieves a practical connexion from inception to finish. All cells are so transferred, in sequence, over this practical connexion. Asynchronous Transportation Manner: The heading includes information about the contents of the warhead and about the method of transmittal. The heading contains merely 5 eights. It was shortened every bit much as possible, incorporating the lower limit reference and command maps for a on the job system. The subdivisions in the heading are a series of spots which are recognized and processed by the ATM bed. Sections included in the heading are Generic Flow Control ( GFC ) , Cell Loss Priority ( CLP ) , Payload Type, Header Error Control, the Virtual Path Identifier and the Virtual Channel Identifier. The Header is the information field that contains the gross bearing warhead. A GFC is a 4-bit field intended to back up simple executions of multiplexing. The GFC is intended to back up flow control. The CLP spot is a 1-bit field that indicates the loss precedence of an single cell. Cells are assigned a binary codification to indicate either high or low precedence. A cell loss precedence value of zero indicates that the cell contents are of high precedence. High precedence cells are least likely to be discarded during periods of congestion. Those cells with a high precedence will merely be discarded after all low precedence cells have been discarded. Cell loss is more damaging to informations transmittal than it is to voice or picture transmittal. Cell loss in informations transmittal consequences in corrupted files. The Payload Type subdivision is a 3-bit field that discriminates between a cell warhead transporting user informations or one transporting direction information. User information is informations of any traffic type that has been packaged into an ATM cell. An illustration of direction Asynchronous Transfer Mode: information is information involved in call set-up. This subdivision besides notes whether the cell experienced congestion. The Header Error Control field consists of mistake look intoing spots. The Header Error Control field is an 8-bit Cyclic Redundancy Code to cheque for individual spot and some multi-bit mistakes. It provides error checking of the heading for usage by the Transmission Convergence ( TC ) sublayer of the Physical bed. The Virtual Path Identifier in the cell heading identifies a package of one or more VCs ( practical channels ) .The Birtual Channel Identifier ( VCI ) in the cel heading identifies a individual VC on a paricular Virtual Path. The way is divided into channels. The pick of the 48 byte warhead was made as a via media to suit multiple signifiers of traffic. The two campaigner warhead sizes were ab initio 32 and 64 bytes. The size of the cell has and consequence on both transmittal efficiency and packetization hold. A long warhead is more efficient than a little warhead since, with a big warhead, more informations can be transmitted per cell with the same sum of operating expense ( heading ) . For informations transmittal entirely, a big warhead is desirable. The longer the warhead is, nevertheless, the more clip is exhausted packaging. Certain traffic types are sensitive to clip such as voice. If packaging clip is excessively long, and the cells are non sent off rapidly, the quality of the voice transmittal will diminish. The 48 byte warhead size was the consequence of a via media that had to be reached between the 64 byte warhead which would supply efficient informations transportation but hapless quality voice and the 32 byte warhead which could Asynchronous Transportation Manner: transmit voice without reverberation but provided inefficient informations transportation. The 48 byte warhead size allows ATM to transport multiple signifiers of traffic. Both time-sensitive traffic ( voice ) and time-insensitive traffic can be carried with the best possible balance between efficiency and packetization hold. ATM Advantages: 1. ATM supports voice, picture and informations leting multimedia and assorted services over a individual web. 2. High development possible, works with bing, bequest engineerings 3. Supply the best multiple service support 4. Supports delay near to that of dedicated services 5. QoS ( Quality of Service ) classes 6. Supply the capableness to back up both connection-oriented and connectionless traffic utilizing AALs ( ATM Adaptation Layers ) 7. Able to utilize all common physical transmittal waies ( DS1, SONET ) 8. Cable can be twisted-pair, coaxal or fiberoptic 9. Ability to link LAN to Wan 10. Bequest LAN emulation 11. Efficient bandwidth usage by statistical multiplexing 12. Scalability 13. Higher sum bandwidth 14. High velocity Mbps and perchance Gbps Asynchronous Transfer Manner: ATM disadvantages: 1. Flexible to efficiency # 8217 ; s disbursal, at present, for any one application it is normally possible to happen a more optimized 2. Technology 3. Cost, although it will diminish with clip 4. New client premises hardware and package are required 5. Competition from other engineerings -100 Mbps FDDI, 100 Mbps Ethernet and fast ethernet 6. Soon the applications that can profit from ATM such as multimedia are rare 7. The delay, with all the promise of ATM # 8217 ; s capablenesss many inside informations are still in the criterions procedure Asynchronous Transfer Mode Bibliography Mention: 1. Freeman, Roger L. ( ( 1996 ) . Telecommunication System Engineering: Third Edition. City: New York, John Wiley A ; Sons, INC. 2. Spohn, Darren L. ( 1997 ) . Data Network Design. City: McGraw-Hill Company. 3. Taylor, D. Edgar ( 1995 ) . The McGraw-Hill Internetworking Handbook. City: New York, McGraw-Hill Company. Internet: 1. Quigley, David ( 1997 ) . A Technical View of ATMs. [ on-line ] , Available: hypertext transfer protocol: //www.mathcs.carleton.edu/students/quigleyd/atmtech.html.

Keine Lazarovitch Essays

Keine Lazarovitch Essays Keine Lazarovitch Essay Keine Lazarovitch Essay Irving Layton was written about his mother, Kline, most likely as a eulogy after she died In 1959. The unusual yet astonishing thing about this poem Is during the first four paragraphs the mood is dark, almost evil like, and fierce, as he speaks of growing old and death, For her mouth was not water but a curse, (paragraph 2). We can see that the speaker, her son, is an honest and expressive man. The emotional effects of these four paragraphs makes us question why Irving would bother to write a eulogy for his mother if he only states readers things about her, it also makes the reader believe perhaps his mother was an unhappy miserable woman who only cursed Gods creatures. The final paragraph of the poem leaves the reader with a satisfying sense of peace, where he basically says that although his mother spoke her mind and was firm on her beliefs, and though the things did may have sometimes been a nuisance, she was his mother. The things she did made her the person she was. Her characteristics were the things that made her real. Though eulogies are usually spoken with a soft tone, and speak of all the great things the person did, the reality Is o one is perfect, and the flaws that people have make them who they are. The authors purpose was to show his true love for his mother. He loved her because she was, in fact, so fierce and outspoken, which is why the thought of this poem is so important. The imagery is very powerful as we can see the picture of his deceased mothers head on the cold pillow, her white watermarking hair in the cheeks hollows, (paragraph 1). This immediately illustrates his mother in the casket at her funeral. Also, we can see the obvious look of the mother when he talks of her amber dads that she wore upon her breast so radiantly. (paragraph 4). These clear visual characteristics also show Layton observance and attention towards his mother. Although this poem does not include a uniform rhyme scheme, we notice the last word of every line (except the last line of each paragraph) has the same ending. For example, in the first paragraph the first line ends with the word pillow, the second line ends with the word hollows and the third line, the word how. Though these words do not necessarily rhyme, they make connections to each other through sound. Also, there are many literary devices in the poem that help emphasize the imagery and sound. For example, in the first paragraph, the alliteration of white watermarking hair really catches the readers attention and creates a greater visual image. Another example would be In the third paragraph when he says Till popularizing Death leaned down and took them for his mould. This line represents growing old and death as a reason for the loss of her rich, black eyebrows. Yet, death cannot physically lean down and take eyebrows, for that is only a human trait, thus making tons Ellen a Tort AT personalization.

Macroeconomics Part 2 Essay Example | Topics and Well Written Essays - 1500 words

Macroeconomics Part 2 - Essay Example Demand pull inflation is caused by a rise in aggregate demand which means persistent rightward shifts in the aggregate demand curve. The rise in aggregate demand may occur due to rises in consumer demand, in the level of government expenditure, in investment by firm, in foreign residents demand for the country’s exports or a combination of these four (Sloman 1997). Demand pull inflation is usually linked to a booming economy. When the economy is in recession, demand pull inflation tends be low. However, when the economy is near the peak of the business cycle, demand pull inflation is likely to be high. The graph above illustrates the rise in aggregate demand by a rightward shift in the aggregate demand curve, from AD1 to AD2. Prices rise from P1 to P2 and output rises from Q1 to Q2 resulting in inflation. On the other hand is the cost push inflation where high costs force firms to increase their prices (Gillespie 2001). Aggregate supply is the total amount of goods and service s produced at a given price level in an economy. When there is a fall in the aggregate supply of goods and services caused by an increase in the cost of production, cost-push inflation occurs.  Cost-push inflation essentially means that prices have rose by an increase in the costs of any of the four factors of production that is; labor, capital, land or entrepreneurship given that firms are already managing at maximum capacity. With increased costs and maximized productivity, firms cannot sustain profit margins by producing the same quantity of goods and services. Consequently, the increased costs are borne by consumers, causing an upward shift in the general price level. The graph above shows the amount of output that can be attained at the given price level.  As production costs escalate, aggregate supply falls from AS1 to AS2 (given production is at maximum capacity), causing the prices to increase from P1 to P2 and total output to decrease from Q1 to Q2. Demand pull and cost push inflation can occur together, since price rises can be caused both by increases in aggregate demand and by independent causes pushing up costs. Similar is the case with the UK’s economy. The UK Consumer Prices Index (CPI) annual inflation rate went up to 4.5% in April, from 4% in March (BBC 2011). As always, there are elements of both types of inflation in the UK’s economy. With the ongoing recovery and a slight increase in demand, there is a small level of demand pull inflation. However, the majority of the effect is cost-push. The increase in VAT is one of the major reasons of inflation in this economy, as well as increases in non-discretionary items such as fuel, utilities, housing and food. These are all necessities whose price hikes act more like an additional tax. The figure below shows the change in the UK’s annual quarterly rate of inflation over the last 15 years. b) Keeping inflation down to a desirable moderate level is an important contributive factor to sustain economic growth. This is because it serves as an incentive for increasing output, investments and unemployment. A rapid rate of inflation disrupts regular economic life leading to a wider income gap, falling output and unemployment. However, the remedy for such inflation depends on the cause. Therefore, government must diagnose its causes before implementing policies. Government policies may pull the rate of inflation down through contractionary fiscal and monetary policies. Monetary policy covers government changes in either the supply of

Employment at Will and Due Process Assignment Example | Topics and Well Written Essays - 1500 words

Employment at Will and Due Process - Assignment Example The greatest intellectual strength is the inclusion of arguments made against their own point. If the authors had failed to include opposing arguments, their article would have been very one-sided and un-credible. It is important for readers to understand both sides of an argument before understanding which side is right or wrong (if there are, in fact, objectively right and wrong sides). This strength of the article, however, also proved to be somewhat of a weakness, because some of the opposing arguments were left unchallenged by the authors. One of the most interesting and perhaps most valid arguments made in this article is that the differences between private and public businesses are becoming less and less clear. Werhane and Radin put forth the notion that public businesses are businesses that cater to the public good before trying to make a profit whereas private businesses function for profit only. While this seems hard to define a business by for legal issues, I have heard that private businesses are marked by having 25 employees or less. I dislike these sort of bright-line policies where a difference of only 1 (say 26 employees instead of 25) makes a tremendous difference in applicable policy. The authors could have used this point to further argue their perspective, but since they did not I will now return to what they did say. Werhane and Radin backed their argument, that the line once drawn between private and public businesses is fading, by a case study involving General Motors (GM). The scenario explained in this article is that the private company GM was declared, by the Supreme Court, able to take over property to expand because it was for the "common good" even though, as a private company, its primary goal is profitability. On the authors' parts, this is a valid argument and it was good to utilize this case study as evidence of their point. (I would have liked more case studies to be used to give solid examples of their arguments.) While the case study does illustrate their point, it is actually not that simple, however. When this happened, it was likely the topic of much subjective debate, because many people may have disagreed with the Supreme Court's ruling. Furthermore, the actual intentions of the Supreme Court may not have been quite so innocent. Corruption is ugly, but it is widespread. Impor tant figures within the Supreme Court could have been easily influenced by a promise of shared wealth from GM. This just goes to show that while the Supreme Court's actions may have led the authors to believe that there is little difference between private and public businesses but really, the ruling of Supreme Court may have been swayed by external factors and its implications are thus inconclusive. Additionally, although Werhane and Radin tried to say that private businesses are like public businesses because they can be deemed as putting efforts towards achieving the common good, I believe it may more often be the other way around. Do public businesses actually put the common good before profitability Without profit, businesses cannot succeed. Perhaps, then, public businesses are similar to private businesses, because they do put profit first.  

Greek and spanish economy over the past three years Essay

Greek and spanish economy over the past three years - Essay Example The year 2007 saw one of the most devastating of all financial crises of all times, which swept over the entire globe. Greece was in a rather juvenile phase during that time as it had not gained ample experience over its past phase of recovery, when the nation had depended substantially on transfer payments from its neighbours. Hence, it was expected that the nation could not avoid a financial crisis. The Ministry of Economy of Greece expected a fall in the annual economic growth rate from 3.6% to 2.4% between 2007 and 2011. Prior to the shock, the nominal economic growth rate in Greece was found to be 4% in the first quarter of 2007. However, given the high rate of inflation integral to that of the nation, the real economic growth rate turned out to be much lower than was officially recorded. The true figures have been presented in the underlying graph. The annual average growth rate, adjusted for inflation, had been recorded at 0.95, 0.18 and -0.65 respectively during 2007 to 2009. These extremely low figures give a hint about the failure of the national government in reviving the economic conditions of Greece. In addition to the poor GDP growth figures, the problems of unemployment and inflation had plagued over the economy since 2007, though improvements have been made in various developmental aspects like those of education, poverty and health. The rate of inflation had reached a peak during 2008, when the average rate had lingered around 4% throughout the year, i.e., by the middle of the term of the newly elected ND government. Though the situation slightly improved by the middle of 2009, it again went unbound by the end of the year (refer to Figure 1.2). Philips curve model of inflation imposes the fact that the rate of inflation prevailing in a nation is inversely related to the rate of unemployment it is experiencing. A similar

Fibonacci Series And The Golden Ratio Engineering Essay

Fibonacci Series And The Golden Ratio Engineering Essay The research question of this extended essay is, Is there a relation between the Fibonacci series and the Golden Ratio? If so be the reason, what is it and explain it. The Fibonacci series, which was first introduced by Leonardo of Pisa (Fibonacci), was found to have had a close connection with the Golden Ratio. The relation found was that the limit of the ratios of the numbers in the Fibonacci sequence converges to the golden mean/golden ratio. I decided to carry out a few set of experiments that involved individual concepts of both: the Fibonacci series and the Golden Ratio. Using their individual applications such as the Golden Rectangle, a computerized calculation supported by a sketched graph, I found that I could arrive at a conjecture that linked the two concepts. I also used the Fibonacci spiral and Golden spiral to find the limit where the values would tend to meet. After carrying out the experiments, I decided to find the proof of the relation using the Binets formula which is essentially the formula for the nth term of a Fibonacci sequence. However, the Binets formula was interesting enough to make me find its proof and solve it myself. From there, I proceeded on to the proof of the relation between the Fibonacci series and the Golden Ratio using this formula. The Binet formula is given by ; . Following the proof, I carried out steps to verify it by substituting different values to check its validity. After proving the validity of the conjecture, I arrived at the conclusion that such a relation does exist. I also learned that this relation had applications in nature, art and architecture. Apart from these, there is a possibility that there are other applications which can be subjected to further investigation. Table of Contents Sl. No. Contents Page No. 1. Introduction to the Fibonacci Series 4 2. Introduction to the Golden Ratio 5 3. The Relationship between them 6 4. Forming the conjecture 6 5. Testing the conjecture 7 6. The proof 15 7. Verification of the proof 20 8. Conclusion 22 9. Further Investigation 22 10. Bibliography 23 Introduction The Fibonacci Series The Fibonacci series is that sequence where every term is the sum of the two terms that precedes it (in the Hindu-Arabic system) where the first two terms of the sequence are 0 and 1. The Fibonacci series is shown below 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 à ¢Ã¢â€š ¬Ã‚ ¦ Where the first two terms are 0 and 1 and the term following it is the sum of the two terms preceding it, which in this case are 0 and 1. Hence, 0 + 1 = 1 (third term) Similarly, Fourth term = third term + second term Fourth term = 1 + 1 = 2 And so the sequence follows. The series was first invented by an Italian by the name of Leonardo Pisano Bigollo (1180 1250) in 1202. He is better known as Fibonacci which essentially means the son of Bonacci. In his book, Liber Arci, there was a puzzle concerning the breeding of rabbits and the solution to this puzzle resulted in the discovery of the Fibonacci series. The problem was based on the total number of rabbits that would be born starting with a pair of rabbits first followed by the breeding of new rabbits which would also start giving birth one month after they were born themselves.  [1]   The problem was broken down into parts and the answer that was obtained gave rise to the Fibonacci series. The Fibonacci series gained a worldwide acceptance soon as after its discovery and was used in many fields. It had its uses and applications in nature (such as the petals of a sunflower and the nautilus shell). Shown below is the application of the series on the whirls of a pine cone.  [2]   http://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fib2.jpghttp://www.3villagecsd.k12.ny.us/wmhs/Departments/Math/OBrien/fib3.jpg The Golden Mean / Golden Ratio The golden mean, also known as the golden ratio, as the name suggests is a ratio of distances in simple geometric figures  [3]  . This is only one of the many definitions found for the term. It is not solely restricted to geometric figures but the proportion is used for art, nature and architecture as well. From pine cones to the paintings of Leonardo Da Vinci, the golden proportion is found almost everywhere. Another definition of the golden ratio is a precise way of dividing a line  [4]   There has never been one concrete definition for the golden ratio which makes it susceptible to different definitions using the same concept. First claimed to be known by Pythagoreans around 500 B.C., the golden proportion was established in print in one of Euclids major works namely, Elements, once and for all in 300 B.C. Euclid, the famous Greek mathematician was the first to establish what the golden section really was with respect to a line. According to him, the division of a line in a mean and extreme ratio  [5]  such a way that the point where this division takes place, the ratio of the parts of the line would be the Golden proportion. He determined that the Golden Ratio was such that The golden ratio is denoted by the Greek alphabet which has a value of 1.6180339à ¢Ã¢â€š ¬Ã‚ ¦ Since then, the golden ratio has been used in various fields. In art, Leonardo Da Vinci coined the ratio as the Divine Proportion and used it to define the fundamental proportions of his famous painting of The Last Supper as well as Mona Lisa. http://goldennumber.net/images/davinciman.gif Finally, it was in the 1900s that the term Phi was coined and used for the first time by an American mathematician Mark Barr who used the Greek letter phi to name this ratio.  [6]  Hence, the term obtained a chain of different names such as the golden mean, golden section and golden ratio as well as the Divine proportion.   The Relation between the Fibonacci series and the Golden Ratio After the discovery of the Fibonacci series and the golden ratio, a relation between the two was established. Whether this relation was a coincidence or not, no one was able to answer this question. However, today, the relation between the two is a very close one and it is visible in various fields. The relation is said to be The limit of the ratios of the numbers in the Fibonacci sequence converges to the golden ratio. This means that as we move to the nth term in the Fibonacci sequence, the ratios of the consecutive terms of the Fibonacci series arrive closer to the value of the golden mean ().  [7]   Forming the Conjecture The Fibonacci series and the golden ratio have been linked together in many ways. Hence, I shall now produce the same statement as a conjecture as I am about to prove the relation through a set of experiments and eventually proving the conjecture (right or wrong). The conjecture is stated below The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ , where n is the nth term of the Fibonacci sequence. In order to prove this conjecture, I have carried out a few experiments below that shall attribute to the result of the above conjecture. Testing the Conjecture Experiment No. 1: The first set of experiments deal with the Golden Rectangle. The golden rectangle is that rectangle whose dimensions are in the ratio (where y is the length of the rectangle and x is the breadth of the rectangle), and when a square of dimensions is removed from the original rectangle, another golden rectangle is left behind. Also, the ratio of the dimensions ( is equal to the golden mean (). I have used the concept of the Golden Rectangle to test whether the ratios of the dimensions of the two golden rectangles, when equated to each other, give the value of the golden ratio or not which is also said to be the formula for the nth term of the Fibonacci series. The latter part of the statement is in accordance with Binets formula. The following experiment shows how this works. Let us consider a rectangle with dimensions . The dotted line is the line that has divided the rectangle in such a way that the square on the left has dimensions of . Now, the rectangle on the right has the dimensions of where x is now the length of the new golden rectangle formed and (y-x) is the breadth. Golden Rectangle 1: y x y-x The reason why this rectangle is called a Golden Rectangle is because the ratio of its dimensions gives the value of à Ã¢â‚¬  . Hence, the information we can gather from the above figure is that (1) The new golden rectangle formed from the above one is shown below with dimensions Golden Rectangle 2: y x x The above new golden rectangle shown must thus also have the same property as that of any other golden rectangle. Therefore, From the above experiments we can establish the following relation (2) For convenience sake, I have decided to take so as to make y the subject of the equation. Hence, the above equation can now be re-written as On cross-multiplying the terms above we get Writing the above equation in the form of a quadratic equation, we get Using the quadratic formula, , we get Hence, the two roots obtained are However, the second root is rejected as a value as y is a dimension of the rectangle and hence cannot be a negative value. Hence we have, Evaluating this value we have But, from equation 1, we know that However, the value of x was restricted to 1 in the above test. So as to eliminate the variable in order to keep only y as the subject, I carried out the calculations below that help in doing so Rewriting the equation Cross-multiplying the variables Dividing the equation by , we get But we know that . Thus, using this substitution in the above equation we have This is the same quadratic that we obtained earlier and hence the doubt for the presence of x clears out. Experiment No. 2: For my second experiment, I have decided to use the concept of the Fibonacci spiral and that of the Golden Spiral. The steps on how to draw these spirals are given below A Fibonacci spiral is formed by drawing squares with dimensions equal to the terms of he Fibonacci series. We start by first drawing a 1 x 1 square 1 x 1 Next, another 1 x 1 square is drawn on the left of the first square. (every new square is bordered in red) Now, a 2 x 2 square is drawn below the two 1 x 1 squares. Next, a 3 x 3 square is drawn to the right of the above figure. Now, a 5 x 5 square is adjoined to the top of the figure. Next, a 8 x 8 square is adjoined to the left of the figure. And so the figure continues in the same manner. The squares are adjoined to the original shape in a left to right spiral (from down to up) and each time the square gets bigger but with dimensions equal to the numbers in the Fibonacci series. Starting from the inner square, a quarter of an arc of a circle is drawn within the square. This step is repeated as we move outward, towards the bigger square. The spiral eventually looks like this http://library.thinkquest.org/27890/media/fibonacciSpiralBoxes.gif The shape shown below is the Fibonacci spiral without the squares http://library.thinkquest.org/27890/media/fibonacciSpiral2.gif A similar process is followed for forming the golden spiral. However, the only difference is that we draw the outer squares first and then draw the arcs starting from the larger squares. Hence, the spiral turns inwards all the way to the inner squares. Golden Spiral The Golden spiral eventually looks like this Golden Spiral On comparing the two spirals, it can be seen that they overlap as the arcs occupy the squares with dimensions of the latter terms of the Fibonacci series. An image of how the two spirals look is shown below http://library.thinkquest.org/27890/media/spirals.gif From the above experiment, it can be seen that there is a connection between the Fibonacci series and the Golden Mean as their individual spirals overlap each other as the n (which is the nth term in the series) tends to infinity. Experiment No. 3: My third experiment involves technology. In this experiment, I decided to use a program of Microsoft Office, namely, Microsoft Excel in order to record the values obtained on calculating the ratio of the consecutive terms of the Fibonacci series. In the table below, I have recorded the terms of the Fibonacci series in the first column, the value of the ratio of the consecutive terms in the Fibonacci sequence in the second column, the value of  [8]  in the third column and the variation of the value of the ration from the value of à Ã¢â‚¬   in the last column. Term of Fibonacci Series Value of ratio of consecutive terms value of variation of value calculated from value of 0 1 1 1.00000000000000 1.61803398874989 0.61803398874989 2 2.00000000000000 1.61803398874989 -0.38196601125011 3 1.50000000000000 1.61803398874989 0.11803398874989 5 1.66666666666667 1.61803398874989 -0.04863267791678 8 1.60000000000000 1.61803398874989 0.01803398874989 13 1.62500000000000 1.61803398874989 -0.00696601125011 21 1.61538461538462 1.61803398874989 0.00264937336527 34 1.61904761904762 1.61803398874989 -0.00101363029773 55 1.61764705882353 1.61803398874989 0.00038692992636 89 1.61818181818182 1.61803398874989 -0.00014782943193 144 1.61797752808989 1.61803398874989 0.00005646066000 233 1.61805555555556 1.61803398874989 -0.00002156680567 377 1.61802575107296 1.61803398874989 0.00000823767693 610 1.61803713527851 1.61803398874989 -0.00000314652862 987 1.61803278688525 1.61803398874989 0.00000120186464 1597 1.61803444782168 1.61803398874989 -0.00000045907179 2584 1.61803381340013 1.61803398874989 0.00000017534976 4181 1.61803405572755 1.61803398874989 -0.00000006697766 6765 1.61803396316671 1.61803398874989 0.00000002558318 10946 1.61803399852180 1.61803398874989 -0.00000000977191 17711 1.61803398501736 1.61803398874989 0.00000000373253 28657 1.61803399017560 1.61803398874989 -0.00000000142571 46368 1.61803398820532 1.61803398874989 0.00000000054457 75025 1.61803398895790 1.61803398874989 -0.00000000020801 121393 1.61803398867044 1.61803398874989 0.00000000007945 196418 1.61803398878024 1.61803398874989 -0.00000000003035 317811 1.61803398873830 1.61803398874989 0.00000000001159 514229 1.61803398875432 1.61803398874989 -0.00000000000443 The aim of the table is to find out whether the value of the ratio reaches the value of à Ã¢â‚¬   or not, as the number of terms increases infinitely. Observation: From the above table, it can be seen that as we reach the nth term of the Fibonacci series, the variation in the value of the ratios from the value of à Ã¢â‚¬  , decreases. This observation is in agreement with the conjecture The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ , where n is the nth term of the Fibonacci sequence. Inference: From the above 3 experiments, I have found that the conjecture holds true for them all. Hence, I would like to state that the tests for the conjectures have been significantly successful. The Proof In order to find the relation between the Fibonacci series and the Golden Ratio, I followed the proof below that uses calculus to establish the required relation. The Fibonacci series is given by, Assuming that 0, 1, and 1 are the first three terms of the sequence: (3) This eventually goes on to form the well known sequence: 0, 1, 1, 2, 3, 5, 8, 13à ¢Ã¢â€š ¬Ã‚ ¦ Dividing the Left Hand Side (or LHS) and the Right Hand Side (or RHS) of equation 3 by F(n), gives (By taking the numerator as the denominator of F(n)) By substituting the limit of the ratios of the terms (as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ ) of the Fibonacci series with A, the limit is taken on both sides such that n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ The above is true as the ratio Hence, the below quadratic equation is formed We can find the roots of A by using the quadratic formula, . or From this we find that This value of is easily attainable using the Binet formula. The Binet formula is that formula which gives the value of by substituting the variable x with one of the n terms of the Fibonacci series. Using the concept of the golden rectangle, the quadratic that was obtained earlier Gave the value of . The proof of the Binet formula shows another possibility to arrive at the relation between the Fibonacci series and the Golden Ratio. The beauty of this proof is that the quadratic first arose from the Fibonacci series calculation and the root that was obtained gave the value of phi. This is from the proof that was written above. Under the heading Testing the Conjecture that was done earlier, the quadratic arose from the dimensions of the Golden Rectangle and the equation thus obtained gave the value of phi. Using this concept, I have followed the proof below which was solved by older mathematicians. The Binet formula is given by Now, from the above tests, we got However, there were 2 values that were obtained on calculating the value of y. The value of y that was negative was rejected then as it was incorrect to consider it a valid answer for a dimension of a geometric figure. Calling this negative root as , we can rewrite the Binet formula as Going back to the quadratic equation, we can substitute in place of y and so the quadratic equation is (4) This quadratic was obtained from the Golden Rectangle. In order to arrive at the Fibonacci sequence, a series of algebraic manipulations will help us reach that step. To start off with, we have the value of in terms of . Now, to get the value of in terms of , we multiply equation (4) into . Using equation (4), we substitute for and we get Using the same method to find the value for raised to higher powers, we have Similarly, Writing the various values for raised to higher powers (5) à ¢Ã¢â€š ¬Ã‚ ¦ Now if we look at the coefficients closely, we see that they are the consecutive terms of the Fibonacci series. This can be written as (6) However, the above trend is not enough proof for generalizing the above statement. Hence, I decided to prove it by using the principle of mathematical induction. Step 1: Step 2: To prove that P(1) is true. Hence, P(1) is true (from equation 5) Step 3: Hence, P(k) is true where Step 4: To prove that P(k+1) is true. Starting from the RHS, (from equation 3) (from equation 4) (from P(k)) = RHS Hence, P(k+1) is true. Therefore, P(n) is true for all Now that we have proved that P(n) is true is true in its generalized form. Also, we know that is the other root of the quadratic equation and so the above general equation can be written in the above form as well (7) In order to obtain the Binet formula in the form of We can subtract equation (7) from equation (6) to get Substituting the original values of and in denominator of the above equation, we get Substituting the value of and in the above equation, we get This is the Binet formula which we started to prove. Hence, the formula is valid. Verifying the Proof In order to validate a proof, it must be tested in order to check whether the conjecture is valid and can be generalized. For this reason, I have decided to use the Binet formula (that was proved above) to check the validity of the relation between the Fibonacci series and the Golden Ratio by substituting values for x in the equation Using Case 1: , Which is the first term of the Fibonacci series. Case 2: , Which is the second term of the Fibonacci series. Case 3: , Which is the third term of the Fibonacci series. Case 4: , Which is the fourth term of the Fibonacci series. From these substitutions it is clear that the formula is a valid one which gives the desired result. Also, the above calculations have proved to be substantial examples for proving the validity of the proofs shown above. However, an important note to remember in the Binet formula is that the value of x starts from 0 and increases. So it can be said that (x belongs to the set of whole numbers). This is to account for the fact that the Fibonacci series starts from 0 and then continues. Hence, the conjecture is true and can be generalized. Hence the conjecture below can be considered true. The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ , where n is the nth term of the Fibonacci sequence. Conclusion From the above tests and verifications, it is clear that a relation between the Fibonacci series and the Golden Ratio does truly exist. The relation being The limit of the ratios of the terms of the Fibonacci series converge to the golden mean as n à ¢Ã¢â‚¬  Ã¢â‚¬â„¢ , where n is the nth term of the Fibonacci sequence. The Fibonacci series as well as the Golden Ratio have their individual applications as well as combined applications in various fields of nature, art, etc. As mentioned earlier, the Fibonacci series was used to find a solution to the rabbit problem. The relation between the two concepts was an integral part of the central idea in the novel The Da Vinci Code. Along with these well known ideas, other applications of the two concepts are present in the whirls of a pine cone, the paintings of Leonardo Da Vinci, the spiral of the nautilus shell, the petals of the sunflower. These are only very few examples regarding the applications of the two concepts. However, this relation has proved to be useful to environmentalists, artists and many other researches. For example, artists were able to use the study of the concept in the paintings of Leonardo Da Vinci and decipher old symbols. It also has given them the chance to create art of their own that by using this concept in their procedure of creating. Further Investigation With the great number of applications that were found regarding the Fibonacci series and the Golden Ratio, there is a possibility that there are other applications of the concept as well. The convergence of the ratios of the values to the value of phi may prove to be of great significance if applied to another theory that has boggled minds of mathematicians for years. Possibilities such as these give rise to the question of further investigation in this aspect of the relationship between the two concepts.

True Happiness in The Sirens of Titan by Kurt Vonnegut and Hans Weingar

True Happiness in The Sirens of Titan by Kurt Vonnegut and Hans Weingartner's The Eduakators A large parcel of the population has as their ultimate goal in life achieving well-being. Unfortunately many try to achieve it through the wrong means. For instance, in The Sirens of Titan, by Kurt Vonnegut, Malachi Constant thinks he is truly happy, but what he really does is fulfill his hedonism, satisfy his shallow needs, without truly searching for a higher form of well-being. Not only does a life focused on hedonic satisfaction not achieve true happiness, it also leads, along with the urge to accumulate, egocentrism, and greed, to an unethical life. The Sirens of Titans depicts this kind of life, which is also represented throughout The Edukators, directed by Hans Weingartner. Both Malachi Constant and Hardenberg believe that money is the solution to all of their problems while ignoring the problems their own lifestyle is causing to other people and society as a whole. Happiness, our own and other people’s, is achieved by focusing our lives in the right things. Even though hedonic satisfaction is necessary for living a happy life, focusing only on hedonic pleasure will have the opposite effect. If you focus on money and the things it can buy as the source for your well-being, you are excluding a series of factors that are necessary to achieve a true state of well-being. The following passage from the article â€Å"On Happiness and Human Potentials: A Review of Research on Hedonic and Eudaimonic Well-Being† clearly details that: Drawing from the eudaimonic view and from SDT, Kasser & Ryan (1993, 1996) related money and materialism to well-being. They predicted that people who place a strong value on wealth... ...se’s life. The only way humanity can achieve true well-being is if abdicates its urge to accumulate and refocus its mostly hedonic ways to a more eudaimonic way of life. Works Cited The Edukators. Dir. Hans Weingartner. IFC Films, 2004. Mill, John Stuart. Utilitarianism. Indianapolis: Hackett Publishing Company, 1979. Nenno, Nancy P. â€Å"Postcards from the Edge.† Light Motives: German Popular Film in Perspective. Eds. Halle, Randall and McCarthy, Margaret. Detroit: Wayne State University Press, 2003. 61-84. Reed, Peter J. "Kurt Vonnegut, Jr." Dictionary of Literary Biography. ed. 1978. Ryan, Richard M., and Edward L. Deci. "On Happiness and Human Potentials: A Review of Research on Hedonic and Eudamonic Well-Being." Annual Review of Psychology 52 (2001): 141-166. Vonnegut, Kurt. The Sirens of Titan. New York: Dell Publishing, 1998.

McNeal Book Review Final Essay

Abstract This paper will constitute a review of Practicing Greatness: 7 Disciplines of Extraordinary Spiritual Leaders,1 with attention given to the disciplines themselves, as well as the rationale and method that McNeal believes will lead to leadership success. The work begins with a quotation from Elton Trueblood that sets the tone for the book’s contents. Trueblood states that â€Å"Deliberate mediocrity is a sin,†2 and to be mediocre is to be without discipline. McNeal penned this work to highlight the disciplines that lead to greatness, both spiritual and in leadership. Interestingly, the listed  Ã¢â‚¬Å"Disciplines† require a course of action on the part of the reader; and this implies not being idle or in the words of Trueblood, mediocre. The â€Å"Disciplines† comprise seven chapters and are noted as follows: The discipline of self—awareness is crucial as it safeguards the leader against unhealthy views of self and needs as well as from task oriented rather than people oriented. The discipline of self—management supports the claim that great leaders are great managers, not merely of others but, primarily and chiefly, of themselves. The discipline of self—development is indicative of all great leaders. They will never stop learning and developing. The discipline of mission honors the propensity of great leaders to sacrifice themselves to great causes. The discipline of decision making sets great leaders apart from good or average leaders. The discipline of belonging characterizes great leaders’ ability to retain and nurture significant relationships that in turn nurture their lives. The discipline of aloneness celebrates great leaders’ ability and grace not only to endure the loneliness of leadership but to actually build solitude into their lives. The over-arching theme of the book, is the spiritual leader that is truly â€Å"great,† achieves that distinction not â€Å"for what they do for themselves or even as a way to become recognized as great leaders. Their end game is about expanding the kingdom of God.†3 Great leaders are cognizant of their inner selves and the signals they send to others via actions. In Boundaries, Cloud and Townsend list four boundary personalities that can derail a leaders’ ability to maintain trust and influence in those they lead. These boundaries are noted as â€Å"Complaints, Avoidants, Controllers and Non-responsives.†4 To augment the above, McNeal cites Gary McIntosh and Samuel Rima’s identification of the leaders’ â€Å"dark-side† comprised of the following characteristics: 1. Compulsive Leaders 2. Narcissistic Leaders 3. Paranoid Leaders 4. Co-dependent Leaders5 McNeal notes that â€Å"Great leaders are great managers—not just managers of projects or other people but mostly of themselves.†6 Yet they are also distributors of â€Å"blessing and encouragement†7 with their work done in humility and in a servant mentality, guaranteeing â€Å"extraordinary character  and exceptional competence developed over time.†8 The author writes with people in view first, and then delves into the varied aspects of leadership based on the disciplines listed in the contents of the book. McNeal draws from years of ministry and teaching experience to demonstrate from Scripture that biblical leadership is possible if one is committed to looking at themselves in light of what Scripture states regarding our condition. Current patterns and preconceptions must be dealt with before change can be implemented; and McNeal provides support from biblical characters who, while not perfect, heeded sound wisdom and learned from experiences so that they would be able to become prepared for what God had planned for them to do. In this regard, McNeal states that all spiritual leaders must flesh out superlatives to distinguish the essence of their call from God to ministry. Questions to be asked in this regard below, will aid the future/current leader in providing answers to questions he/she might have regarding their present ministry or avocation: a. What people or cause do you feel drawn to? b. What do you want to help people do or achieve or experience? c. How do you want to help people? d. What message do you want to deliver? e. How do you intend to serve or have an impact on the world? f. Why did you say yes to God to begin with?9 Mc Neal expounds on leadership and those who will seek to carry it out. the work is not overtly religious, yet it is balanced in the biblical references included. The illustrations of real people in real situations and with real leadership styles are instrumental in bringing clarity and focus to an exhaustive subject. The author has clearly demonstrated his objectives set out in the introduction, and has provided examples for leadership that are able to be implemented in all business applications and not merely the church only. This work is to be commended for anyone interested in not only what makes leaders great; but as well, how they arrived at the summit and are able to remain there. Two things are clear from a complete reading of this book: 1. Great spiritual leaders are committed consciously and intentionally to the spiritual disciplines 2. Great leaders feel blessed, have an attitude of gratitude and have chosen excellence before God and men. Response One major life experience that this book triggered involves the section of â€Å"Managing Expectations in The Discipline of Self-Management.10 I had recently been promoted to assistant manager at my place of employment where I was to be responsible for the implementation of new sales protocols. In reading McNeal, and in retrospect, I realize that because an understanding of self-awareness was lacking, I set expectations so high my natural and learned abilities could not stay even with them. I failed in goals I set and therefore lost confidence in my ability to manage others who worked under me. I knew that there were things which were wrong in how I was doing things; yet I could not figure it out. I arrived at the point where I felt that I would become ill anytime I had to make decisions on the job. I sat down and cried because things seemed to have no solution where it seemed, I was able to find solutions and fix things. I remember hearing a preacher once who was teaching on the wisdom of God and the finiteness of the mind of men. I took my Bible out and went to the concordance where I searched for words and phrases relating to wisdom, mind and knowledge, and I was led to Proverbs 3:5-6 which states to 5 Trust in the Lord with all thine heart; and lean not unto thine own understanding. 6 In all thy ways acknowledge him, and he shall direct thy paths (KJV). After studying these verses, I realized that not only was I lacking understanding of self-awareness, I was lacking in acknowledging God faithfully considering his infinite wisdom and sovereignty. This was the point where I had to confess my sins of ingratitude and ignorance of God and his power and wisdom. I knew in my heart that I would have to pray and listen to God through his Word more than I ever had; and I knew that I would have to be disciplined so that I would not find it easy to revert to where I had been before in my working life without him. Reflection One question that immediately came to my mind the further I went in this book was why McNeal did not incorporate more Scripture references than he did, or at the least alluded to? The â€Å"disciplines† of extraordinary spiritual leaders, one might think, would be found in Scripture with an excursus into what these disciplines entail. Were the decision left to me, I would have  drawn especially from the teachings of Christ; and from various leaders found within the pages of the sacred text.11 In retrospect, McNeal gave considerable attention to various disciplines within the teaching (illustrating) and ministry (practical) of Christ; yet the reader would likely desire more from the author in these regards than what he did present. There were areas of this work that read more as a psychological development course than the dynamics of spiritual development as the sanctified life of the leader will become apparent within his or her duties regardless the arena they work in. In r eading and discussing this book with my husband, I feel that a sense of balance would have been achieved were McNeal delved a bit more into biblical application of the topics he presented throughout. In terms of fleshing out the ordinary from the extraordinary, McNeal provides generous circumstances and situations from his own ministry life to demonstrate that every aspect of self-awareness and development hinge upon how the person views him/her-self in light of the truth. These â€Å"truths† are the non-negotiable prima fasciae of obedience to God and his will. In terms of readability, this work does not pose difficulty in determining where the author is headed in his teaching. The main issue is that more references to biblical characters might help to balance the illustrations of modern day people within various ministry or organizational structures. Action One of the first things I aim to accomplish in my life is to focus more on God and his wisdom rather than my own. It is so easy and tempting to second guess what one should do to achieve desired results; and more often than not, I have been guilty of over-guessing what I should do to the point that I am correcting every aspect of something to the point of micro-management and monarchial temperament. In the second place, I must set aside daily and consistent times to be alone with God in prayer and meditation on him rather than myself and my needs. I realize that most issues may be solved with remembering that â€Å"he must increase while I decrease† (Jn. 3:30). The power of God is not going to be neither availed nor prevalent if one does not fully relinquish the reins of their life to him, thus following rather than leading him. The above can have no time-table for measurement, so it seems best to state that it is a daily discipline that only grows and develops  properly over a course of time never ending. My ministry now and in the future will very likely utilize vast sections of this work with a focus on the three â€Å"Self’s:† Self-Awareness, Self-Management and Self-Development. I must commit to long-term developed and sustained growth interspersed with bench-marks as a measurement to demonstrate that I am growing and ministering properly. The people I will eventually teach and lead have a right to know what will be expected of them; they also have a right to point out the missteps leaders can make. Here is where I need to be receptive to criticism and rebuke; not wearing my emotions on my sleeve, rather, considering what is being said and then praying to God for the mind to take the necessary steps to corrective action and further development. I know where I am at now, even if I have not fully figured out everything about myself. I do anticipate a long road ahead toward restructuring and complete discipline; yet I believe that â€Å"the race does not belong to the swift, but to those who will never quit† (Eccl. 9:11). Bibliography McNeal, Reggie. Practicing Greatness: 7 Disciplines of Extraordinary Spiritual Leaders. San Francisco, CA: Jossey-Bass, 2006.

Sequences on ACT Math Strategy Guide and Review

Sequences on ACT Math Strategy Guide and Review SAT / ACT Prep Online Guides and Tips Sequences are patterns of numbers that follow a particular set of rules. Whether new term in the sequence is found by an arithmetic constant or found by a ratio, each new number is found by a certain rule- the same rule- each time. There are several different ways to find the answers to the typical sequence questions- †What is the first term of the sequence?†, â€Å"What is the last term?†, â€Å"What is the sum of all the terms?†- and each has its benefits and drawbacks. We will go through each method, and the pros and cons of each, to help you find the right balance between memorization, longhand work, and time strategies. This will be your complete guide to ACT sequence problems- the various types of sequences there are, the typical sequence questions you’ll see on the ACT, and the best ways to solve these types of problems for your particular ACT test taking strategies. Before We Begin Take note that sequence problems are rare on the ACT, never appearing more than once per test. In fact, sequence questions do not even appear on every ACT, but instead show up approximately once every second or third test. What does this mean for you? Because you may not see a sequence at all when you go to take your test, make sure you prioritize your ACT math study time accordingly and save this guide for later studying. Only once you feel you have a solid handle on the more common types of math topics on the test- triangles (comng soon!), integers, ratios, angles, and slopes- should you turn your attention to the less common ACT math topics like sequences. Now let's talk definitions. What Are Sequences? For the purposes of the ACT, you will deal with two different types of sequences- arithmetic and geometric. An arithmetic sequence is a sequence in which each term is found by adding or subtracting the same value. The difference between each term- found by subtracting any two pairs of neighboring terms- is called $d$, the common difference. -5, -1, 3, 7, 11, 15†¦ is an arithmetic sequence with a common difference of 4. We can find the $d$ by subtracting any two pairs of numbers in the sequence- it doesn’t matter which pair we choose, so long as the numbers are next to one another. $-1 - -5 = 4$ $3 - -1 = 4$ $7 - 3 = 4$ And so on. 12.75, 9.5, 6.25, 3, -0.25... is an arithmetic sequence in which the common difference is -3.25. We can find this $d$ by again subtracting pairs of numbers in the sequence. $9.5 - 12.75 = -3.25$ $6.25 - 9.5 = -3.25$ And so on. A geometric sequence is a sequence of numbers in which each successive term is found by multiplying or dividing by the same amount each time. The difference between each term- found by dividing any neighboring pair of terms- is called $r$, the common ratio. 212, -106, 53, -26.5, 13.25†¦ is a geometric sequence in which the common ratio is $-{1/2}$. We can find the $r$ by dividing any pair of numbers in the sequence, so long as they are next to one another. ${-106}/212 = -{1/2}$ $53/{-106} = -{1/2}$ ${-26.5}/53 = -{1/2}$ And so on. Though sequence formulas are useful, they are not strictly necessary. Let's look at why. Sequence Formulas Because sequences are so regular, there are a few formulas we can use to find various pieces of them, such as the first term, the nth term, or the sum of all our terms. Do take note that there are pros and cons for memorizing formulas. Pros- formulas are a quick way to find your answers, without having to write out the full sequence by hand or spend your limited test-taking time tallying your numbers. Cons- it can be easy to remember a formula incorrectly, which would lead you to a wrong answer. It also is an expense of brainpower to memorize formulas that you may or may not even need come test day. If you are someone who prefers to use and memorize formulas, definitely go ahead and learn these! But if are not, then you are still in luck; most (though not all) ACT sequence problems can be solved longhand. So if you have the patience- and the time to spare- then don’t worry about memorizing formulas. That all being said, let’s take a look at our formulas so that those of you who want to memorize them can do so and so that those of you who don’t can still understand how they work. Arithmetic Sequence Formulas $$a_n = a_1 + (n - 1)d$$ $$\Sum \terms = (n/2)(a_1 + a_n)$$ These are our two important arithmetic sequence formulas and we will go through how each one works and when to use them. Terms Formula $a_n = a_1 + (n - 1)d$ If you need to find any individual piece of your arithmetic sequence, you can use this formula. First, let us talk about why it works and then we can look at some problems in action. $a_1$ is the first term in our sequence. Though the sequence can go on infinitely, we will always have a starting point at our first term. $a_n$ represents any missing term we want to isolate. For instance, this could be the 4th term, the 58th, or the 202nd. Why does this formula work? Well let’s say we wanted to find the 2nd term in the sequence. We find each new term by adding our common difference, or $d$, so the second term would be: $a_2 = a_1 + d$ And we would then find the 3rd term in the sequence by adding another $d$ to our existing $a_2$. So our 3rd term would be: $a_3 = (a_1 + d) + d$ Or, in other words: $a_3 = a_1 + 2d$ And the 4th term of the sequence, found by adding another $d$ to our existing third term, would continue this pattern: $a_4 = (a_1 + 2d) + d$ Or $a_4 = a_1 + 3d$ So, as you can see, each term in the sequence is found by adding the first term to $d$, multiplied by $n - 1$. (The 3rd term is $2d$, the 4th term is $3d$, etc.) So now that we know why the formula works, let’s look at it in action. What is the difference between each term in an arithmetic sequence, if the first term of the sequence is -6 and the 12th term is 126? 3 4 6 10 12 Now, there are two ways to solve this problem- using the formula, or finding the difference and dividing by the number of terms between each number. Let’s look at both methods. Method 1: Arithmetic Sequence Formula If we use our formula for arithmetic sequences, we can find our $d$. So let us simply plug in our numbers for $a_1$ and $a_n$. $a_n = a_1 + (n - 1)d$ $126 = -6 + (12 - 1)d$ $126 = -6 + 11d$ $132 = 11d$ $d = 12$ Our final answer is E, 12. Method 2: finding difference and dividing Because the difference between each term is regular, we can find that difference by finding the difference between our terms and then dividing by the number of terms in between them. Note: be very careful when you do this! Though we are trying to find the 12th term, there are NOT 12 terms between the first term and the 12th- there are actually 11. Why? Let’s look at a smaller scale sequence of 3 terms. 4, __, 8 If you wanted to find the difference between these terms, you would again find the difference between 4 and 8 and divide by the number of terms separating them. You can see that there are 3 total terms, but 2 terms separating 4 and 8. 1st: 4 to __ 2nd: __ to 8 When given $n$ terms, there will always be $n - 1$ terms between the first number and the last. So, if we turn back to our problem, now we know that our first term is -6 and our 12th is 126. That is a difference of: $126 - -6$ $126 + 6$ $132$ And we must divide this number by the number of terms between them, which in this case is 11. $132/11$ $12$ Again, the difference between each number is E, 12. As you can see, the second method is just another way of using the formula without actually having to memorize the formula. How you solve these types of questions completely depends on how you like to work and your own personal ACT math strategies. Sum Formula $\Sum \terms = (n/2)(a_1 + a_n)$ This formula tells us the sum of the terms in an arithmetic sequence, from the first term ($a_1$) to the nth term ($a_n$). Basically, we are multiplying the number of terms, $n$, by the average of the first term and the nth term. Why does this work? Well let’s look at an arithmetic sequence in action: 4, 7, 10, 13, 16, 19 This is an arithmetic sequence with a common difference, $d$, of 3. A neat trick you can do with any arithmetic sequence is to take the sum of the pairs of terms, starting from the outsides in. Each pair will have the same exact sum. So you can see that the sum of the sequence is $23 * 3 = 69$. In other words, we are taking the sum of our first term and our nth term (in this case, 19 is our 6th term) and multiplying it by half of $n$ (in this case $6/2 = 3$). Another way to think of it is to take the average of our first and nth terms- ${4 + 19}/2 = 11.5$ and then multiply that value by the number of terms in the sequence- $11.5 * 6 = 69$. Either way, you are using the same basic formula, so it just depends on how you like to think of it. Whether you prefer $(n/2)(a_1 + a_n)$ or $n({a_1 + a_n}/2)$ is completely up to you. Now let’s look at the formula in action. Andrea is selling boxes of cookies door-to-door. On her first day, she sells 12 boxes of cookies, and she intends to sell 5 more boxes per day than on the day previous. If she meets her goal and sells boxes of cookies for a total of 10 days, how many boxes total did she sell? 314 345 415 474 505 As with almost all sequence questions on the ACT, we have the choice to use our formulas or do the problem longhand. Let’s try both ways. Method 1: formulas We know that our formula for arithmetic sequence sums is: $\Sum = (n/2)(a_1 + a_n)$ In order to plug in our necessary numbers, we must find the value of our $a_n$. Once again, we can do this via our first formula, or we can find it by hand. As we are already using formulas, let us use our first formula. $a_n = a_1 + (n - 1)d$ We are told that the first term in our sequence is 12. We also know that she sells cookies for 10 days and that, each day, she sells 5 more boxes of cookies. This means we have all our pieces to complete this formula. $a_n = 12 + (10 - 1)5$ $a_10 = 12 + (9)5$ $a_10 = 12 + 45$ $a_10 = 57$ Now that we have our value for $a_n$ (in this case $a_10$), we can complete our sum formula. $(n/2)(a_1 + a_n)$ $(10/2)(12 + 57)$ $5(69)$ $345$ Our final answer is B, 345. Method 2: longhand Alternatively, we can solve this problem by doing it longhand. It will take a little longer, but this way also carries less risk of mis-remembering a formula. The decision is, as always, completely up to you on how you choose to solve these kinds of questions. First, let us write out our sequence, beginning with 12 and adding 5 to each subsequence number, until we find our nth (10th) term. 12, 17, 22, 27, 32, 37, 42, 47, 52, 57 Now, we can either add them up all by hand- $12 + 17 + 22 + 27 + 32 + 37 + 42 + 47 + 52 + 57 = 345$ Or we can use our arithmetic sequence sum trick and divide the sequence into pairs. We can see that there are 5 pairs of 69, so $5 * 69 = 345$. Again, our final answer is B, 345. Whoo! Only one more formula to go! Geometric Sequence Formulas $$a_n = a_1( r^{n - 1})$$ (Note: there is a formula to find the sum of a geometric sequence, but you will never be asked to find this on the ACT, and so it is not included in this guide.) This formula, as with the first arithmetic sequence formula, will help you find any number of missing pieces in your sequence. Given two pieces of information about your sequence ($a_n$ $a_1$, $a_1$ $r$, or $a_n$ $r$), you can find the third. And, as always with sequences, you have the choice of whether to solve your problem longhand or with a formula. What is the first term in a geometric sequence if each number is found by multiplying the previous term by -3 and the 8th term is 4,374? -0.222 0.667 -2 6 -18 Method 1: formula If you’re one for memorizing formulas, we can simply plug in our values into our equation in place of $a_n$, $n$, and $r$ in order to solve for $a_1$. $a_n = a_1( r^{n - 1})$ $4374 = a_1(-3^{8 - 1})$ $4374 = a_1(-3^7)$ $4374 = a_1(-2187)$ $-2 = a_1$ So our first term in the sequence is -2. Our final answer is C, -2. Method 2: longhand Alternatively, as always, we can take a little longer and solve them problem by hand. First, set out our number of terms in order to keep track of them, with our 8th term, 4374, last. ___, ___, ___, ___, ___, ___, ___, 4374 Now, let’s divide each number by -3 down the sequence until we reach the beginning. ___, ___, ___, ___, ___, ___, -1458, 4374 ___, ___, ___, ___, ___, 486, -1458, 4374 And, if we keep going thusly, we will eventually get: -2, 6, -18, 54, -162, 486, -1458, 4374 Which means that we can see that our first term is -2. Again, our final answer is C, -2. As with all sequence solving methods, there are benefits and drawbacks to solving the question in each way. If you choose to use formulas, make very sure you can remember them exactly. And if you solve the questions by hand, be very careful to find the exact number of terms in the sequence. The ACT will always provide bait answers for anyone who is one or two terms off the nth term- in this problem, if you had accidentally assigned 4374 as the 7th term or the 9th term, you would have chosen answer B or D. Once you find the strategy that works best for you, the pieces will all fall into place. Typical ACT Sequences Questions Because all sequence questions on the ACT can be solved (if sometimes arduously) without the use or knowledge of sequence formulas, the test-makers will only ever ask you for a limited number of terms or the sum of a small number of terms (usually less than 12). As we saw above, you may be asked to find the 1st term in a sequence, the nth term, the difference between your terms (whether a common difference, $d$, or a common ratio, $r$), or the sum of your terms in arithmetic sequences only. You also may be asked to find an unusual twist on a sequence question that combines your knowledge of sequences. For example: What is the sum of the first 5 terms of an arithmetic sequence in which the 6th term is 14 and the 11th term is 22? 2.2 6.0 12.4 32.6 46.0 Again, let us look at both formulaic and longhand methods for how to solve a problem like this. Method 1: formulas In order to find our common difference, we can use our main arithmetic sequence formula. But this time, instead of beginning with the actual $a_1$, we are beginning with our 6th term, as this is what we are given. Essentially, we are designating our 6th term as our 1st term and our 11th term as our 6th term and then plugging these values into our formula. $a_n = a_1 + (n - 1)d$ $22 = 14 + (6 - 1)d$ $22 = 14 + 5d$ $8 = 5d$ $1.6 = d$ Now, we can find our actual 1st term by using the $d$ we just found and our 11th term value of 22. $a_n = a_1 + (n - 1)d$ $22 = a_1 + (11 - 1)1.6$ $22 = a_1 + (10)1.6$ $22 = a_1 + 16$ $6 = a_1$ The 1st term of our sequence is 6. Now, we need to find the 5th term of our sequence in order to use our arithmetic sequence sum formula to find the sum of the first 5 terms. $a_n = a_1 + (n - 1)d$ $a_5 = 6 + (5 - 1)1.6$ $a_5 = 6 + (4)1.6$ $a_5 = 6 + 6.4$ $a_5 = 12.4$ And finally, we can find the sum of our first 5 terms by using our sum formula and plugging in the values we found. $(n/2)(a_1 + a_n)$ $5/2(6 + 12.4)$ $2.5(18.4)$ $46$ Our final answer is E, 46. As you can see, this problem still took a significant amount of time using our formulas because there were so many moving pieces. Let us look at this problem were we to solve it longhand instead. Method 2: longhand First, let us find our common difference by finding the difference between our 6th term and our 11th term and dividing by how many terms are in between them, which in this case is 5. (Why 5? There is one term between the 6th and 7th terms, another between the 7th and 8th, another between the 8th and 9th, another between the 9th and 10th, and the last between the 10th and 11th terms. This makes a total of 5 terms.) This gives us: $22 - 14 = 8$ $8/5 = 1.6$ Now, let us simply find all the numbers in our sequence by working backwards and subtracting 1.6 from each term. ___, ___, ___, ___, ___, 14, ___, ___, ___, ___, 22 ___, ___, ___, ___, ___, 14, ___, ___, ___, 20.4, 22 ___, ___, ___, ___, ___, 14, ___, ___, 18.8, 20.4, 22 And so on, until all the spaces are filled. 6, 7.6, 9.2, 10.8, 12.4, 14, 15. 6, 17.2, 18.8, 20.4, 22 Now, simply add up the first 5 terms. $6 + 7.6 + 9.2 + 10.8 + 12.4$ $46$ Our final answer is E, 46. Again, you always have the choice to use formulas or longhand to solve these questions and how you prioritize your time (and/or how careful you are with your calculations) will ultimately decide which method you use. You've seen the typical ACT sequence questions, so let's talk strategies. Tips For Solving Sequence Questions Sequence questions can be somewhat tricky and arduous to slog through, so keep in mind these ACT math tips on sequences as you go through your studies: 1: Decide before test day whether or not you will use the sequence formulas Before you go through the effort of committing your formulas to memory, think about the kind of test-taker you are. If you are someone who lives and breathes formulas, then go ahead and memorize them now. Most sequence questions (though, as we saw above, not all of them) will go much faster once you have the formulas down straight. If, however, you would rather dedicate your time and brainpower to other math topics or to the method of performing sequence questions longhand, then don’t worry about your formulas! Don’t even bother to try to remember them- just decide here and now not to use them and forget about the formulas entirely. Unless you can be sure to remember them correctly, a formula will hinder more than help you when it comes time to take your ACT, so make the decision now to either memorize them or forget about them. 2: Write your values down and keep your work organized Though many calculators can perform long strings of calculations, sequence questions by definition involve many different values and terms. Small errors in your work can cause a cascade effect. One mistyped digit in your calculator can throw off your work completely, and you won’t know where the error happened if you do not keep track of your values. Always remember to write down your values and label them in order to prevent a misstep somewhere down the line. 3: Keep careful track of your timing No matter how you solve a sequence question, these types of problems will generally take you more time than other math questions on the ACT. For this reason, most all sequence questions are located in the last third of the ACT math section, which means the test-makers think of sequences as a â€Å"high difficulty† level problem. Time is your most valuable asset on the ACT, so always make sure you are using yours wisely. If you can answer two other math questions in the time it takes you to answer one sequence question, then maximize your point gain by focusing on the other two questions. Always remember that each question on the ACT math section is worth the same amount of points, so prioritize quantity and don’t let your time run out trying to solve one problem. If you feel that you can answer a sequence problem quickly, go ahead! But if you feel it will take up too much time, move on and come back to it later. Ready to put your knowledge to the test? Test Your Knowledge Now let’s test your sequence knowledge with real ACT math problems. 1. What is the first term in the arithmetic sequence if terms 6 through 9 are shown below? ...196, 210, 224, 238 7 14 98 126 140 2. What is the sum of the first 8 terms in the arithmetic sequence that begins: 7, 10.5, 14,... 143.5 154 162.5 168 176.5 3. Answers: D, B, E Answer Explanations: 1. As always, we can solve this problem with formulas or via longhand. For the sake of brevity, we will only use one method per problem here. In this case, let us solve our problem via longhand. We are told this is an arithmetic sequence, so we can find our common difference by subtracting neighboring terms. Let us take a pair and subtract to find our $d$. $238 - 224 = 14$ $d = 14$ We know our common difference is 14, and 196 is our 6th term. Let us work backwards to find our 1st term. ___, ___, ___, ___, ___, 196, 210, 224, 238 ___, ___, ___, ___, 182, 196, 210, 224, 238 ___, ___, ___, 168, 182, 196, 210, 224, 238 And so on, until we reach our first term. 126, 140, 154, 168, 182, 196, 210, 224, 238 As long as we kept our work organized, we will find the first term in our sequence. In this case, it is 126. Our final answer is D, 126. 2. Again, we have many options for solving our problem. In this case, we can use a combination of longhand and formula (in addition to the standard options of using either method alone). First, we must find our common difference between our terms by subtracting any neighboring pair. $14 - 10.5 = 3.5$ $d = 3.5$ Now that we have found our $d$, let us finish our sequence until the 8th term by continuing to add 3.5 to each successive term. 7, 10.5, 14, 17.5, 21, 24.5, 28, 31.5 And finally, we can plug in our values into our sum formula to find the sum of all our terms. $(n/2)(a_1 + a_n)$ $(8/2)(7 + 31.5)$ $(4)(38.5)$ $154$ The sum of the first 8 terms in the sequence is 154. Our final answer is B, 154. 3. Again, we can use multiple methods to solve our problem. In this case, let us use our formula for geometric sequences. First, we need to find our common ratio between terms, so let us divide any pair of neighboring terms to find our $r$. ${-27}/9 = -3$ $r = -3$ Now we can plug in our values into our formula. $a_n = a_1( r^{n - 1})$ $a_7 = 1(-3^{7 - 1})$ $a_7 = 1(-3^6)$ $a_7 = 1(-729)$ $a_7 = 729$ The 7th term of our sequence is 729. Our final answer is E, 729. You did it, you genius you! The Take Aways Sequence questions often take a little time and effort to get through, but they are usually made complicated by their number of terms and values rather than being actually difficult to solve. Just remember to keep all your work organized and decide before test-day whether you want to spend your study efforts memorizing, or if you would prefer to work out your sequence problems by hand. As long as you keep your values straight (and don’t get tricked by bait answers!), you will be able to grind through these problems without fail, using either method. What’s Next? Phew! You have officially mastered ACT sequence questions. So...now what? Well you're in luck because there are a lot more ACT math topics and guides to check out! Want to brush up on your ratios? How about your trigonometry? Coordinate geometry and slopes? No matter what ACT topic you want to master, we've got you covered. Feel like you're running out of time on ACT math? Check out our guide to help you beat the clock. Want to know the score you should aim for? Start by looking at how the scoring works and what that means for you. Looking to get a perfect score? Our guide to getting a 36 on ACT math (written by a perfect-scorer) will help you get to where you want to be! Want to improve your ACT score by 4 points? Check out our best-in-class online ACT prep program. We guarantee your money back if you don't improve your ACT score by 4 points or more. Our program is entirely online, and it customizes what you study to your strengths and weaknesses. If you liked this Math lesson, you'll love our program. Along with more detailed lessons, you'll get thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step program to follow so you'll never be confused about what to study next. Check out our 5-day free trial: