The formula

$D(1r{)}^{n}P(\frac{1-(1r{)}^n}{r})$

where

$D$ is the the amount borrowed in dollars

$r$ is the interest rate per period

$P$ is the payment per period in dollars

$n,$ is the number of periods passed since $D$ was borrowed.

will give the amount owed if $$D$ is borrowed at an interest rate of $r($per period$)$ if $$P$ is paid per pay period after $n$ periods have passed. A period could be a month, year or any period of time.

$($a$)$ Suppose $$100$ is borrowed at an interest rate of $\mathrm{.}05$ and $$6$ is paid per period, how much is owed after $8$ periods?

$($b$)($no work required in this part$)$ It can be shown that

$D(1r{)}^{n}P(\frac{1-(1r{)}^n}{r})$

can be rewritten to look like

$(D-\frac{P}{r})(1r{)}^n\frac{P}{r}$

For a challenge you can try this.

$($c$)$ Set $D=$$$100,r=\mathrm{.}05,P=6$ and compute

$\underset{n}{lim}(D-\frac{P}{r})(1r{)}^n\frac{P}{r}$

How should you interpret this in terms of debt.

$($d$)$ Set $D=$$$100,r=\mathrm{.}05,P=4$ and compute

$\underset{n}{lim}(D-\frac{P}{\mathrm{.}05})(1r{)}^n\frac{P}{r}$

How should you interpret this in terms of debt.

$($e$)$ Set $D=$$$100,r=\mathrm{.}05,P=5$ and compute

$\underset{n}{lim}(D-\frac{P}{r})(1r{)}^n\frac{P}{r}$

How should you interpret this in terms of debt. Discuss how this differs from part $($a$).$