Assuming that all payments were the same amount, a payment consists of its interest part and its principal part thus:

This payment schedule assumes that the current payment *x* includes
interest on all of the remaining principal, including principal which is
part of the current payment. Therefore, the first payment includes an interest
payment on all of the borrowed principal.

The final payment looks like the following:

The *Pj*'s may be re-written into a recurrence relation such that:

In order to solve for *x*, one more statement must be made.

OK, now for the hard part. We replace *Ij* of equation (3) using
the relationship given by equation (1) and then substitute the recurrence
identity of equation (2):

We can rewrite the limits of the summation now:

Thanks to equation 3.2.13 on page 227 of *Discrete Mathematics for
Computer Scientists* by Joe Mott, Abe Kandel, and Ted Baker, I can rewrite
the summation in another form, and readjust everything so that:

At last, we have the analytic solution! *Quod erat demonstrandum*
(``That which was to be shown''), otherwise known as Q.E.D.