1044 CHAPTER 11 Further Topics in Algebra The sum of the terms of a sequence, called a series, is written using summation notation. The Greek capital letter sigma g is used to indicate a sum. Series A finite series is an expression of the form Sn =a1 +a2 +a3 +P+ an = a n i=1 ai . An infinite series is an expression of the form SH =a1 +a2 +a3 +P+ an +P= aH i=1 ai . The letter i is the index of summation. CAUTION Do not confuse this use of i with the use of i as the imaginary unit. Other letters, such as k and j, may be used for the index of summation. EXAMPLE 4 Using Summation Notation Evaluate a 6 k=112k + 12. ALGEBRAIC SOLUTION Write each of the six terms, and then evaluate the sum. a6 k=112k + 12 = 121 + 12 + 122 + 12 + 123 + 12 + 124 + 12 + 125 + 12 + 126 + 12 = 12 + 12 + 14 + 12 + 18 + 12 + 116 + 12 + 132 + 12 + 164 + 12 = 3 + 5 + 9 + 17 + 33 + 65 = 132 GRAPHING CALCULATOR SOLUTION A graphing calculator can list the terms of a sequence and then compute the sum of the terms. See Figure 6. Figure 6 S Now Try Exercise 51. EXAMPLE 5 Using Summation Notation with Subscripts Write the terms for each series and, if possible, evaluate each sum. (a) a 6 j=3 aj (b) a 3 i=1 16xi - 22, if x1 = 2, x2 = 4, and x3 = 6 (c) a 4 i=1 ƒ1xi2Δx, if ƒ1x2 = x 2, x 1 = 0, x2 = 2, x3 = 4, x4 = 6, and Δx = 2 SOLUTION (a) a 6 j=3 aj = a3 + a4 + a5 + a6 Replace j in aj with 3, 4, 5, and 6.
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