CHAPTER 11 Further Topics in Algebra New Symbols an nth term of a sequence an i=1 ai summation notation; sum of n terms i index of summation Sn sum of first n terms of a sequence Greek letter sigma aH i=1 ai sum of an infinite number of terms lim nuH Sn limit of Sn as n increases without bound n! n-factorial nCr , C1n, r2 , or An r B binomial coefficient (combinations of n elements taken r at a time) nPr or P1n, r2 permutations of n elements taken r at a time n1E2 number of outcomes that belong to event E P1E2 probability of event E E′ complement of event E Quick Review Concepts Examples The sequence 1, 1 2 , 1 3 , 1 4 , c , 1 n has general term an = 1 n . The corresponding series is the sum 1 + 1 2 + 1 3 + 1 4 + g+ 1 n . Evaluate each series. The arithmetic sequence 2, 5, 8, 11, chas a1 = 2. d = 5 - 2 = 3 Common difference (Any two successive terms could have been used.) a6 i=1 5 = 6 # 5 = 30 a4 i=1 312i + 12 = 3 a 4 i=1 12i + 12 = 313 + 5 + 7 + 92 = 72 a3 i=11 5i + 6i22 = a 3 i=1 5i + a 3 i=1 6i2 = 15 + 10 + 152 + 16 + 24 + 542 = 30 + 84 = 114 1112 11.1 Sequences and Series A finite sequence is a function that has a set of natural numbers of the form 51, 2, 3, c , n6 as its domain. An infinite sequence has the set of natural numbers as its domain. The nth term of a sequence is symbolized an. A series is an indicated sum of the terms of a sequence. Summation Properties If a1, a2, a3, c , an and b1, b2, b3, c , bn are two sequences and c is a constant, then for every positive integer n, the following hold true. (a) a n i=1 c =nc (b) a n i=1 cai =c a n i=1 ai (c) a n i=1 1 ai tbi 2 = a n i=1 ai t a n i=1 bi Summation Rules an i=1 i =1 +2 +P+n = n1n +12 2 an i=1 i2 =12 +22 + P+n2 = n1n +12 12n +12 6 an i=1 i3 =13 +23 + P+n3 = n21n +122 4 11.2 Arithmetic Sequences and Series Assume a1 is the first term, an is the nth term, and d is the common difference in an arithmetic sequence. Common Difference d =an+1 −an
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