Level Payment Problem (C)

One important application of formula for the sum of a geometric progression is the Level Payment Problem. You borrow A dollars for N periods at a rate of r per period. What should your payment P per period be?

Observe that at the end of the first period, when you make your payment you will owe

A(1+r) - P

since your new balance was A+rA and you paid off P of it. At the end of the second period you will owe

(A(1+r) - P)(1+r) - P = A(1+r)² - P((1+r) + 1).

At the end of the third period you will owe

(A(1+r)² - P((1+r) + 1))(1+r) - P = A(1+r)³ - P((1+r)² + (1+r) + 1).

Continuing in this fashion, we see that at the end of the $N^{\rm th}$ period you owe

$$A(1+r)^N - P((1+r)^{N-1} + \cdots + (1+r) +1) = A(1+r)^N - P\frac{(1+r)^N-1}{r}$$

if $r\neq 0$ and

A - NP

if r = 0. If you are to owe zero after the $N^{\rm th}$ payment then

$$P = A\frac{r(1+r)^N}{(r+1)^N-1} = \frac{Ar}{1-(1+r)^{-N}}$$

if $r\neq 0$ and

$$P = \frac{A}{N}$$

if r = 0.