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.