An annuity is a sequence of equal payments that are paidor received at regular time

Chapter 4, Problem 37

(choose chapter or problem)

An annuity is a sequence of equal payments that are paid or received at regular time intervals. For example, you may want to deposit equal amounts at the end of each year into an interest-bearing account for the purpose of accumulating a lump sum at some future time. If, at the end of each year, interest of \(i \times 100 \%\) on the account balance for that year is added to the account, then the account is said to pay \(i \times 100 \%\) interest, compounded annually. It can be shown that if payments of Q dollars are deposited at the end of each year into an account that pays \(i \times 100 \%\) compounded annually, then at the time when the nth payment and the accrued interest for the past year are deposited, the amount \(S(n)\) in the account is given by the formula

\(S(n)=\frac{Q}{i}\left[(1+i)^{n}-1\right]\)

Suppose that you can invest $5000 in an interest-bearing account at the end of each year, and your objective is to have $250,000 on the 25 th payment. Approximately what annual compound interest rate must the account pay for you to achieve your goal? [Hint: Show that the interest rate i satisfies the equation \(50 i=(1+i)^{25}-1\), and solve it using Newton's Method.]

Equation Transcription:

Text Transcription:

i x 100%

S(n)

S(n)=Q/i[(1+i)^n -1]

50i=(1+i)^25 -1