## Thinking Mathematically (6th Edition)

$PMT = \frac{P~(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}]}$
This is the formula we use to calculate the value of an annuity: $A = \frac{PMT~[(1+\frac{r}{n})^{nt}-1]}{\frac{r}{n}}$ Note that $PMT$ is the monthly payment. This is the formula we use when we make calculations with compound interest: $A = P~(1+\frac{r}{n})^{nt}$ Note that $P$ is the amount of the loan. We can equate $A$ in both equations and solve for $PMT$ $\frac{PMT~[(1+\frac{r}{n})^{nt}-1]}{\frac{r}{n}} = P~(1+\frac{r}{n})^{nt}$ $PMT~[(1+\frac{r}{n})^{nt}-1] = P~(1+\frac{r}{n})^{nt}~(\frac{r}{n})$ $PMT = \frac{P~(1+\frac{r}{n})^{nt}~(\frac{r}{n})}{[(1+\frac{r}{n})^{nt}-1]}$ $PMT = \frac{P~(\frac{r}{n})}{\frac{[(1+\frac{r}{n})^{nt}-1]}{(1+\frac{r}{n})^{nt}}}$ $PMT = \frac{P~(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}]}$