1066 CHAPTER 11 Further Topics in Algebra Sum of theTerms of an Infinite Geometric Sequence The sum S∞ of the terms of an infinite geometric sequence with first term a1 and common ratio r, where 0 r 0 61, is given by the following. SH = a1 1 −r If r 71, then the terms increase without bound in absolute value, so there is no limit as nS∞. Therefore, if ∣ r∣ +1, then the terms of the sequence will not have a sum. EXAMPLE 8 Evaluating Infinite Geometric Series Evaluate each sum. (a) a ∞ i=1 a- 3 4b a- 1 2b i-1 (b) a ∞ i=1 a 3 5b i SOLUTION (a) Here, a1 = - 3 4 and r = - 1 2 . Because 0 r 0 61, the sum converges. S∞ = a1 1 - r = - 3 4 1 - A - 1 2B = - 3 4 3 2 = - 3 4 , 3 2 = - 3 4 # 2 3 = - 1 2 (b) a ∞ i=1 a 3 5b i = 3 5 1 - 3 5 = 3 5 2 5 = 3 5 , 2 5 = 3 5 # 5 2 = 3 2 a1 = 3 5 , r = 3 5 S Now Try Exercises 61 and 63. EXAMPLE 9 Finding the Future Value of an Annuity To save money for a trip, Jacqui deposited $1000 at the end of each year for 4 yr in an account paying 2% interest, compounded annually. Find the future value of this annuity. SOLUTION We use the formula for interest compounded annually. A = P11 + r2t The first payment earns interest for 3 yr, the second payment for 2 yr, and the third payment for 1 yr. The last payment earns no interest. 100011.0223 + 100011.0222 + 100011.022 + 1000 Total amount This is the sum of the terms of a geometric sequence with first term (starting at the end of the sum as written above) a1 = 1000 and common ratio r = 1.02. Annuities A sequence of equal payments made after equal periods of time, such as car payments or house payments, is an annuity. If the payments are accumulated in an account (with no withdrawals), the sum of the payments and interest on the payments is the future value of the annuity.
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