You borrow a principal P at monthly interest r (e.g. in a 6% mortgage, r = 0.5% or 0.005). Every month your debt increases by a factor 1+r and you pay X. Let Pn be your debt after the nth month. (a) Show that Po = P and Pn = (1+r) Pn-1-X for n > 1. Unfold the recursion, make a guess for Pn and prove it. (b) Show that the monthly payment X = rP/(1 (1 + r)) will payoff your mortgage after N payments. (c) What monthly payment X pays off a $300,000 mortgage in 15 years, for monthly interest (i) 0.5% (ii) 0.75%?