{"id":5732,"date":"2013-04-16T13:04:07","date_gmt":"2013-04-16T13:04:07","guid":{"rendered":"https:\/\/schooltutoring.com\/help\/?p=5732"},"modified":"2014-12-02T08:27:03","modified_gmt":"2014-12-02T08:27:03","slug":"mathematical-sequences-and-pascals-triangle","status":"publish","type":"post","link":"https:\/\/schooltutoring.com\/help\/mathematical-sequences-and-pascals-triangle\/","title":{"rendered":"Mathematical Sequences and Pascal’s Triangle"},"content":{"rendered":"

Overview:\u00a0 What Are Mathematical Sequences?<\/strong>
\nA mathematical sequence is a function that follows a specific rule and a specific pattern.\u00a0 For example, a number set A consists of {3, 5, 7, 9, 11 …}.\u00a0 Such a number set is easily recognized as the set of odd numbers.\u00a0 If there is no last term, the sequence is infinite, and if there is a last term, the sequence is finite.<\/p>\n

Finding the Rule<\/strong>
\nIn order to find the rule it follows, pair each number with a positive integer first,\u00a0 and then find the nth term in the sequence.\u00a0 Continuing with the example, the first term in the series is 3, so pair (1,3).\u00a0 The second is (2,5); the third is (3,7);\u00a0 the fourth is (4, 9); the fifth is (5, 11).\u00a0 The nth term defines the rule: an<\/sub>= 2n +1.\u00a0 Check to see if the rule is a true expression.\u00a0 Does the sixth term, a6<\/sub>, follow the rule?\u00a0 If so, the sixth pair will be (6, [2(6) +1] ) or (6, 13).<\/p>\n

Arithmetic Sequences<\/strong>
\nAn arithmetic sequence is a sequence in which the difference between successive terms is a constant, called the common difference or d.\u00a0 For example, a2<\/sub> – a1<\/sub>=d and a2<\/sub> = a1<\/sub> +d.\u00a0 Generally, an<\/sub>=a1<\/sub> +(n-1)d.\u00a0 This means that if the first term of an arithmetic sequence is 3 and the difference is 2, the 20th term will be\u00a0 a20<\/sub> = 3 + (20 -1)2, or 3 +38 = 41.<\/p>\n

Geometric Sequences<\/strong>
\nA geometric sequence is a sequence in which the ratio of successive terms is a constant, called the common ratio.\u00a0\u00a0\u00a0 For example, an +1<\/sub>\/an<\/sub> is the same ratio, no matter where the points next to each other are in the sequence.\u00a0 Using the geometric sequence, {4, 8, 16, 32…}, it can be shown that 8\/4 =16\/8 =32\/16, and so on.\u00a0 Generally, an<\/sub>=a(rn-1<\/sup>), where a without the subscript is equal to the first term in the sequence and r is the geometric constant.\u00a0 Therefore, a8<\/sub> = 4(27<\/sup>) or 4(128), or 512.<\/p>\n

Pascal’s Triangle:\u00a0 An Application of Sequences<\/strong>
\nOne of the most famous sequences in mathematics is known as “Pascal’s Triangle”, named after the mathematician who summarized its properties.\u00a0 It is a sequence of binomial coefficients, arranged so that the each number in the triangle is the sum of the two that are above it.\u00a0 The properties of these sequences form the arrangements in probability theory.<\/p>\n

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Overview:\u00a0 What Are Mathematical Sequences? A mathematical sequence is a function that follows a specific rule and a specific pattern.\u00a0 For example, a number set A consists of {3, 5, 7, 9, 11 …}.\u00a0 Such a number set is easily recognized as the set of odd numbers.\u00a0 If there is no last term, the sequence […]<\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[2],"tags":[137,731,1090,1290],"class_list":["post-5732","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-arithmetic-sequences","tag-geometric-sequences","tag-mathematical-sequences","tag-pascals-triangle"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/5732","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=5732"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/5732\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=5732"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=5732"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=5732"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}