Solved: Annuities When an employee receives a paycheck at the end of each month, dollars

Chapter 9, Problem 84

(choose chapter or problem)

Using a Geometric Series In Exercises 83-86, use the formula for the nth partial sum of a geometric series

\(\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}\).

Annuities When an employee receives a paycheck at the end of each month, P dollars is invested in a retirement account. These deposits are made each month for t years and the account earns interest at the annual percentage rate r. When the interest is compounded monthly, the amount A in the account at the end of t years is

\(\begin{aligned} A &=P+P\left(1+\frac{r}{12}\right)+\cdots+P\left(1+\frac{r}{12}\right)^{12 t-1} \\ &=P\left(\frac{12}{r}\right)\left[\left(1+\frac{r}{12}\right)^{12 t}-1\right] \end{aligned} \)

When the interest is compounded continuously, the amount A in the account after t years is

\(\begin{aligned} A &=P+P e^{r / 12}+P e^{2 r / 12}+P e^{(12 t-1) r / 12} \\

&=\frac{P\left(e^{r t}-1\right)}{e^{r / 12}-1} \end{aligned} \)

Verify the formulas for the sums given above.

Text Transcription:

Sum_i=0^n-1 ar^i = a(1-r^n)/1-r

A=P+P(1+r/12)+...+P(1+r/12)^12t-1

=P(12/r)[(1+r/12)^12t - ]

A=P + Pe^r/12 + Pe^2r/12 + Pe^(12t-1)r/12

=P(e^rt-1)/e^r/12-1