1053 11.2 Arithmetic Sequences and Series EXAMPLE 4 FindingTerms of an Arithmetic Sequence Determine an and a18 for the arithmetic sequence having a2 = 9 and a3 = 15. SOLUTION Subtract the given consecutive terms to obtain d = 15 - 9 = 6. The first term, a1, is found as follows. a2 = a1 + d Definition of arithmetic sequence 9 = a1 + 6 Let a2 = 9 and d = 6. a1 = 3 Subtract 6 and interchange sides. Find an by substituting values for a1 and d in the formula for an. an = a1 + 1n - 12d Formula for an an = 3 + 1n - 126 Let a1 = 3 and d = 6. an = 3 + 6n - 6 Distributive property an = 6n - 3 Simplify. Now, find a18. a18 = 61182 - 3 Let n = 18. a18 = 105 Multiply, and then subtract. S Now Try Exercise 27. EXAMPLE 3 FindingTerms of an Arithmetic Sequence Determine a13 and an for the arithmetic sequence -3, 1, 5, 9, c . SOLUTION Here a1 = -3 and d = 1 - 1-32 = 4. To find a13, substitute 13 for n in the formula for the nth term. an = a1 + 1n - 12d Formula for an a13 = -3 + 113 - 124 Let a1 = -3, n = 13, and d = 4. a13 = -3 + 11224 Subtract. a13 = -3 + 48 Multiply. a13 = 45 Add. Find an by substituting values for a1 and d in the formula for an. an = a1 + 1n - 12d Formula for an an = -3 + 1n - 124 Let a1 = -3 and d = 4. an = -3 + 4n - 4 Distributive property an = 4n - 7 Simplify. S Now Try Exercise 23. EXAMPLE 5 Finding the FirstTerm of an Arithmetic Sequence An arithmetic sequence has a8 = -16 and a16 = -40. Determine a1. SOLUTION We obtain a16 by adding the common difference to a8 eight times. a16 = a8 + 8d Definition of arithmetic sequence -40 = -16 + 8d Let a16 = -40 and a8 = -16. -24 = 8d Add 16. d = -3 Divide by 8 and interchange sides.
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