SECTION 12.3 Geometric Sequences; Geometric Series 881 (b) The length of the arc of the nth swing is 0.98 18. n 1 ( ) ⋅ − For the length of the arc to be exactly 12 inches requires that n n 0.98 18 12 0.98 12 18 2 3 1 log 2 3 1 ln 2 3 ln 0.98 1 20.07 21.07 n n 1 1 0.98( ) ( ) ( ) ( ) ⋅ = = = − = = + ≈ + = − − The length of the arc of the pendulum exceeds 12 inches on the 21st swing and is first less than 12 inches on the 22nd swing. (c) After 15 swings, the total distance swung is L 18 0.98 18 0.98 18 0.98 18 0.98 18 2 3 14 ( ) ( ) ( ) = + ⋅ + ⋅ + ⋅ + + ⋅ 1st 2nd 3rd 4th 15th This is the sum of a geometric sequence. The common ratio is 0.98; the first term is 18. The sum has 15 terms, so L 18 1 0.98 1 0.98 18 13.07 235.3 inches 15 = ⋅ − − ≈ ⋅ ≈ The pendulum has swung approximately 235.3 inches after 15 swings. (d) When the pendulum stops, it has swung the total distance T 18 0.98 18 0.98 18 0.98 18 2 3 ( ) ( ) = + ⋅ + ⋅ + ⋅ + This is the sum of an infinite geometric series.The common ratio is r 0.98; = the first term is a 18. 1 = Since r 1, < the series converges. Its sum is T a r 1 18 1 0.98 900 1 = − = − = The pendulum has swung a total of 900 inches when it finally stops. Divide both sides by 18. Express as a logarithm. Solve for n; use the Change-of-Base Formula. Now Work PROBLEM 87 5 Solve Annuity Problems Using Formulas In Section 12.1, we discussed annuity problems using recursive formulas. We now develop a formula that allows us to determine the amount of an ordinary annuity by looking at the sum of a geometric sequence. Recall, when deposits are made at the same time that the interest is credited, the annuity is called ordinary and the amount of an annuity is the sum of all deposits made plus all interest paid. We again discuss only ordinary annuities here. Suppose that the interest rate that an account earns is i percent per payment period (expressed as a decimal). For example, if an account pays 12% compounded monthly (12 times a year), then i 0.12 12 0.01. = = If an account pays 8% compounded quarterly (4 times a year), then i 0.08 4 0.02. = = To develop a formula for the amount of an annuity, suppose that $P is deposited each payment period for n payment periods in an account that earns i percent per payment period. When the last deposit is made at the nth payment period, the first deposit of $P has earned interest compounded for n 1 − payment periods, the second deposit of $P has earned interest compounded for n 2 − payment periods, and so on. Table 8 shows the value of each deposit after n deposits have been made. Deposit 1 2 3 . . . − n 1 n Amount ( ) + − P i 1 n 1 ( ) + − P i 1 n 2 ( ) + − P i 1 n 3 . . . ( ) + P i 1 P Table 8
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