The mortgage calculation we dial in class can also be found as exercise 3.9.18 in Optimal, Integral, Likely. 2. In class, when we computed a mortgage, we set P(n) to be the amount owed on the loan after n months, and approximated the nth payment to principal as P'(n). At that point in the course, we didn't have the tools to compute geometric sums. Now that we do, we can reinvestigate our mortgage model. Let $P be the initial loan, and let T be the monthly interest rate. Suppose you pay off your loan over the course of N months. (a) Interpret the system of equations below in terms of the model. TP-i-k1 =1"(Pk1)+k2 =T(Pk1k2)+k3 =T(Pk1k2"'kn_1)+kn =T(Pk1k2"'kn1kn)+kn+1 Your answer should include the interpretation of the two terms added in each line, and an expla- nation of why the lines are all equal. (b) Find k+1 in terms of kn and 1', using the system of equations in (a). (Assume 1 5 n + 1 S N.) (c) Find 16,, in terms of r and h, using your answer from (b). (Assume n S N.) N (d) Interpret Z k,- in terms of the model. i=1 (9) Find the total monthly payment amount (interest plus principal) in terms of P, N, and 'r. N 1 TN+1 Hint: in small class 7, you saw a formula for geometric sums: Z r" = 11:!) 1/4 (f) In class, we set r = W, P = 750, 000, and N = 300. i. With these values of 1', P, and N , what is the estimated monthly payment? Use a. calculator to evaluate your nal answer, rounding to the nearest cent. 3. Once again, suppose you take out a loan of P dollars, with monthly interest rate r. Suppose also that your bank gives you the option to choose what you pay to principal each month. You expect that your income will increase at a monthly rate of s, from a combination of cost-of-living adjustments and seniority pay. You want to pay o your bank loan by making monthly payments (interest plus principal] that increase in line with your overall income. That is, if you pay $M in interest and principal one month, the next month you'll pay 3M {1 + s) in principal and interest. (a) Set up a dierential equation for P(n}. the amount you owe at months after your loan is taken out. Your equation should include M, s, r, P, P', and n. The differential equation in (a) is not separable, so we haven't given you a way to solve it from scratch. (If you continue learning about differential equations. you'll probably learn how. Alternately, if you get used to computer algebra systems, solving the DE is the part you can leave to a computer coming up with the DE in the rst place is where a human is most needed.) Using sums is pretty tricky here. so both methods we've talked about so far (di'erential equations and geometric sums) come up short. So. as a third option for understanding this loan, let's use a spreadsheet to investigate one particular case. 1/4 1/3 S P = 750,000, - , d - uppose r 100 an 3100. (b) First we'll want to nd M, the total payment to the bank in the rst month. We'll set up a spreadsheet to keep track of things. A template is below. An arrow indicates that the contents of a cell have been copy-pasted down the row. You want to pay off your loan in 300 months. Al-Dl are descriptions of the contents of their columns. You'll be guessing different values of M , and storing them in cell E2. You'll want to know how much debt is left after 300 months, so to save on scrolling, H300) is displayed in cell F2. A B C D E F Inn nn- i. What should you write in cells B3, C3, and D3? For this part, you do not have to justify your answer. ii. You want column D to compute up to P(300). If you copy-paste row 3 down columns B, C, and D, what cell will P(300} be Computed in? (To save on scrolling, write \"=1" into cell F2, where X is the cell where P(3DD) is computed.) For this part, you do not have to justify your answer. iii. Play around with different values of M to make P{300) as close as possible to 0. To within one cent, what is the best value of M? For this part, you do not have to justify your answer. iv. Explain Why M