THEOREM OBJECTIVES Recognize patterns in binomial expansions. Evaluate a binomial coefficient. Expand a binomial raised to a power. Find a particular term in a binomial expansion Understand the principle of mathematical induction. Prove statements using mathematical induction. Definition: BINOMIAL THEOREM Patterns in Binomial Expansions A number of patterns‚ as follows‚ begin to appear when we write the binomial expansion of a b n‚ where n is a positive integer. a
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The United States serves as a giant puzzle. From coast to coast‚ each region of land is slowly being put together. This is very similar to westward expansion. From 1801 to1844‚ westward expansion spread rapidly and America experienced a serious change in national unity. During this period territorial expansion had a very large impact and overall brought Americans together. During the early 1800’s (1800-1823) the Louisiana Purchase and Treaty of 1818 increased national unity. The Louisiana Purchase
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Territorial Expansion Between 1800 and 1850‚ the United States was a nation extending in all possible ways. America experienced a amount change in national unity. Territorial expansion destroyed national unity. Between 1819 and 1824 is when the unified nation of America began to divide and depart from one another. Triggered by the concept of Manifest Destiny‚ people believed that America should extend from “sea to shining sea”. It was the issues of the expansion of slavery into the new territories
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westward expansion was the key to the nation’s health: He believed that a republic depended on an independent‚ virtuous citizenry for its survival‚ and that independence and virtue went hand in hand with land ownership‚ especially the ownership of small farms. (“Those who labor in the earth‚” he wrote‚ “are the chosen people of God.”) In order to provide enough land to sustain this ideal population of virtuous yeomen‚ the United States would have to continue to expand.The westward expansion of the
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COLONIAL EXPANSIONS AND RIVALRIES What is it? Colonialism expansion is where countries (such as Britain‚ France‚ Germany and Russia) settle on new land. They do this to expand their territories‚ and to use their resources for to benefit their empire. Rivalries between the empires are consequential. This is because many empires are involved in conflicts when attempting to regain new land. How did it contribute to the start of World War I? Imperialism was one of the causes of World War I.
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The westward expansion‚ which started in the 1820s‚ was one of the biggest steps forward our country has ever taken. It nearly tripled the size of our country and increased the population by monstrous amounts. It also improved the economy greatly though agricultural means‚ prompted political leaders to sent people to explore (Lewis and Clark)‚ started a civil war‚ motivated huge amounts of people to move west‚ and almost destroyed our country. The expansion was magnified greatly by the
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Purpose: In this lab‚ we will heat up a metal rod and measure how much it expands. We will calculate its Coefficient of Linear Expansion‚ α‚ and prove the formula for thermal expansion‚ ΔL = αLiΔT. Also‚ we heat up a known volume of methyl alcohol and calculate its Coefficient of Volume Expansion‚ β‚ and prove the formula for volume expansion of a liquid ΔV = βViΔT. Equipment: Hollow Metal Rods of different metals‚ Steam Generator‚ Stand‚ Burner‚ Flexible Hoses‚ Motion Amplifier‚ Micrometer
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The period from 1450 to 1650 is often referred to as the “Age of Discovery‚” an era of advances in geographical knowledge and technology. It was also a time of European migration to other parts of the world. This Age of Expansion and European exploration from 1450 to 1525‚ greatly promoted by economical‚ technological‚ political‚ and religious factors‚ launched a major turning point in world history. The economy of late fifteenth century played an important role in European conquest. The Ottoman
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then exercised today. As time progressed‚ the role and influence of the federal government became more prominent. Examples of expansion in federal authority are evident in events surrounding the Civil War‚ Reconstruction‚ the Progressive Era‚ the Great Depression‚ and the Civil Rights Era. Identify and describe four (4) examples of U.S. federal government expansion of authority between the beginning of the U.S. Civil War and the end of the Civil Right Era. These examples must be placed
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Thermal Expansion happens a lot in everyday life. When something is heated and expands this is Thermal Expansion. The way Thermal Expansion works‚ is when it is heated the atoms expand‚ and then when it is cooled it shrinks. Several examples of Thermal Expansi Thermal Expansion happens a lot in everyday life. When something is heated and expands this is Thermal Expansion. The way Thermal Expansion works‚ is when it is heated the atoms expand‚ and then when it is cooled it shrinks. Several
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