454 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions Compound Interest Recall the formula for simple interest, I = Prt, where P is principal (amount deposited), r is annual rate of interest expressed as a decimal, and t is time in years that the principal earns interest. Suppose t = 1 yr. Then at the end of the year, the amount has grown to the following. P + Pr = P11 + r2 Original principal plus interest If this balance earns interest at the same interest rate for another year, the balance at the end of that year will increase as follows. 3P11 + r24 + 3P11 + r24r = 3P11 + r2411 + r2 Factor. = P11 + r22 a # a = a2 After the third year, the balance will grow in a similar pattern. 3P11 + r224 + 3P11 + r224r = 3P11 + r22411 + r2 Factor. = P11 + r23 a2 # a = a3 Continuing in this way produces a formula for interest compounded annually. A =P11 +r2t The general formula for compound interest can be derived in the same way. Compound Interest If P dollars are deposited in an account paying an annual rate of interest r compounded (paid) n times per year, then after t years the account will contain A dollars, according to the following formula. A =P a1 + r nb tn EXAMPLE 7 Using the Compound Interest Formula Suppose $1000 is deposited in an account paying 4% interest per year compounded quarterly (four times per year). (a) Find the amount in the account after 10 yr with no withdrawals. (b) How much interest is earned over the 10-yr period? SOLUTION (a) A = P a1 + r nb t n Compound interest formula A = 1000a1 + 0.04 4 b 10142 Let P = 1000, r = 0.04, n = 4, and t = 10. A = 100011 + 0.01240 Simplify. A = 1488.86 Round to the nearest cent. Thus, $1488.86 is in the account after 10 yr. (b) The interest earned for that period is $1488.86 - $1000 = $488.86. S Now Try Exercise 97(a).
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