Chapter 5.7, Problem 26ES

A person saving for retirement makes an initial deposit of $1,000 to a bank account earning interest at a rate of 3% per year compounded monthly, and each month she adds an additional $200 to the account.
a. For each nonnegative integer n, let $A_n$ be the amount in the account at the end of n months. Find a recurrence relation relating $A_k$ to $A_{k-1}$ .
b. Use iteration to find an explicit formula for $A_n$ .
c. Use mathematical induction to prove the correctness of the formula you obtained in part (b).
d. How much will the account be worth at the end of 20 years? At the end of 40 years?
e. In how many years will the account be worth $10,000’?