1056 CHAPTER 11 Further Topics in Algebra For example, in the sequence of interest payments discussed earlier, n = 30, a1 = 30, and an = 1. Sn = n 2 1a1 + an2 First formula for Sn S30 = 30 2 130 + 12 Let n = 30, a1 = 30, and an = 1. S30 = 151312 Simplify. S30 = 465 Multiply. A total of $465 interest will be paid over the 30 months. EXAMPLE 7 Using the Sum Formulas Consider the arithmetic sequence -9, -5, -1, 3, 7, c . (a) Evaluate S12. (b) Evaluate the sum of the first 60 positive integers. SOLUTION (a) We want the sum of the first 12 terms. Sn = n 2 32a1 + 1n - 12d4 Second formula for Sn S12 = 12 2 321-92 + 112 - 1244 Let n = 12, a1 = -9, and d = 4. S12 = 156 Evaluate. (b) The first 60 positive integers form the arithmetic sequence 1, 2, 3, 4, c , 60. Sn = n 2 1a1 + an2 First formula for Sn S60 = 60 2 11 + 602 Let n = 60, a1 = 1, and a60 = 60. S60 = 1830 Evaluate. S Now Try Exercises 45 and 55. EXAMPLE 8 Using the Sum Formulas The sum of the first 17 terms of an arithmetic sequence is 187. If a17 = -13, find a1 and d. SOLUTION Sn = n 2 1a1 + an2 Use the first formula for Sn. S17 = 17 2 1a1 + a172 Let n = 17. 187 = 17 2 1a1 - 132 Let S17 = 187 and a17 = -13. 22 = a1 - 13 Multiply by 2 17 . a1 = 35 Add 13 and interchange sides.
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