Deriving the Amortization Equation?

Question

Recently, I was given the task of thinking up the Amortized simple interest home loan equation in my head. I couldn't manage to find it, and ended up looking around the net to find that the equation can be found algebraically - as far as I could tell. The people providing the proofs may have skipped some steps when bringing the original equations into the picture.

For some reason, this whole problem brings integration to mind. Can the equation be found SOLELY with algebra, or is calculus necessary somewhere in the proof?

Answer 1

Solely with algebra, alg 2 level. You find the sum of a geometric series after factoring out the (unknown) monthly payment.