Chapter 8.3, Problem 85E

An investor deposits $P$ dollars on the first day of each month in an account with an annual interest rate $r$, compounded monthly. The balance $A$ after $t$ years is

$A=P\left(1+\frac{r}{12}\right)+\mathrm{.........}+P{\left(1+\frac{r}{12}\right)}^{12r}$

Show that the balance is

$A=P\left[{\left(1+\frac{r}{12}\right)}^{12r}-1\right]\left(1+\frac{12}{r}\right)$