In the periodic payment formula - we looked at the future value of each payment. It was impossible to use for long time periods since there were as many terms as compounding periods. So we stepped back, shifted our view of it by transforming it by a common piece (1+rate) and then we combined the shifted equation with the original. The overlap cancelled to give us a general formula that was easy to use and only had a couple of terms. Combining the shifted equation with the original was similar to Jeff Weeks, since each involved looking at a change in view and organization that eventually simplified an impossible problem and made it possible to solve.