1. Computing mortgage payments and balances Suppose you borrow & dollars. which you are to repay over a period of 20 years at 6% interest, compounded monthly. How much will your monthly payments be? To answer that, let p be the monthly payment and A, be the balance remaining on the loan at the end of & months. (a) The key formula is A& = 1.005A&-1 - P. (27) Explain in words what this says and where it comes from. (b) Fork = 1, we have to take Ax-1 = Lin (27). Applying formula (27) recursively for k = 1, 2, 3 gives Al = 1.005L - p A2 = 1.005A, -p = (1.005)24 - 1.005p - p As = 1.005A2 -p = (1.005)34 - (1.005) p - 1.005p - p. Continuing in this way, write a formula for Ax in terms of L and p. (c) To simplify your formula, use the following formula, known as the geometric series formula: For any number & # 1 and any positive integer m, 1+1+1+. ..+ ( =- ( - 1 * - 1 (d) If the loan is to be paid off in 20 years, we need A240 = 0. This leads to an equation in L and p, and solving for p gives you the size of your monthly payment. (e) If you borrow $120,000 to buy a house, to be paid off in 20 years at 6% interest, compounded monthly, what will your monthly payment be? (f) Rework the basic formula to cover the case of interest rate r, compounded n times a year, with the loan to be paid off in Y years. (g) Turn the formula around by solving for L as a function of p to find out how much you may borrow if you can afford to pay p dollars per month. (h) Suppose you want to buy a house, with the loan to be paid off in 30 years at 5% interest, compounded monthly. If you can afford to pay $800 per month, what size loan can you take