882 CHAPTER 12 Sequences; Induction; the Binomial Theorem The amount A of the annuity is the sum of the amounts shown in Table 8; that is, A P i P i P i P P i i 1 1 1 1 1 1 n n n 1 2 1 [ ] ( ) ( ) ( ) ( ) ( ) = ⋅ + + ⋅ + + + ⋅ + + = + + + + + − − − The expression in brackets is the sum of a geometric sequence with n terms and a common ratio of i 1 . ( ) + As a result, A P i i i P i i P i i P i i 1 1 1 1 1 1 1 1 1 1 1 1 n n n n n 2 1 [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + + + + + + + = − + − + = − + − = + − − − The following theorem has been proved: THEOREM Amount of an Annuity Suppose that P is the deposit in dollars made at the end of each payment period for an annuity paying i percent interest per payment period. The amount A of the annuity after n deposits is A P i i 1 1 n ( ) = + − (9) NOTE In formula (9), remember that when the n th deposit is made, the first deposit has earned interest for n 1 − compounding periods and the n th deposit has earned no interest. j Determining the Amount of an Annuity To save for retirement, Miguel decides to place $300 into an individual retirement account (IRA) each quarter (every three months) for the next 40 years. What will the value of the IRA be when Miguel makes his 40th deposit? Assume that the rate of return of the IRA is 7% per annum compounded annually. (This is the historical rate of return in the stock market.) EXAMPLE 10 Solution Because the deposits are made quarterly, there are 4 deposits made per year. This is an ordinary annuity with n 4 40 160 ( ) = = quarterly deposits of P $300. = The rate of interest per payment period is i 0.07 4 0.0175. = = The amount A of the annuity after 160 deposits is A $300 1 0.0175 1 0.0175 $300 860.0672 $258,020.16 160 ( ) = ⋅ + − ≈ ⋅ ≈ Determining the Amount of an Annuity To save for her daughter’s college education, Miranda decides to put $100 aside every month in a credit union account paying 2% interest compounded monthly. She begins this savings program when her daughter is 3 years old. How much will she have saved by the time she makes the 180th deposit? How old is her daughter at this time? EXAMPLE 11 Solution This is an annuity with P $100, = n 180, = and i 0.02 12 . = The amount A of the annuity after 180 deposits is A $100 1 0.02 12 1 0.02 12 $100 209.71306 $20,971.31 180 ( ) = + − ≈ ⋅ ≈ Because there are 12 deposits per year, when the 180th deposit is made 180 12 15 years = have passed, and Miranda’s daughter is 18 years old. Now Work PROBLEM 91
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