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Problem 3: (22 points) Suppose that you obtain a loan of P dollars (the principal value) to purchase a home. The home mortgage has an annual fixed interest rate of R% and a repayment period of M months. 1. (6 points) Let x(k) represent the remaining loan amount after the kth month, and let b represent the monthly payment. Show that x satisfies the first-order difference equation x(k + 1) = ax(k) - b, and specify the value of a in terms of R (note that R% = R/100). 2. (8 points) Using the initial condition x(0) = P and the Z-transform method, solve for x(k) for k ? 0. Express your answer in terms of P, a, and b. 3. (3 points) Using x(M) = 0 and your result from part 2, solve for the monthly payment b. 4. (3 points) Compute the monthly payment for a 3.6% fixed rate (a = 1.003), 30 year loan with principal amount of $225,000 (you may verify your answer using an on-line mortgage calculator). 5. (2 points) Determine the total amount of interest paid for the loan in part 3.

          Problem 3: (22 points)
Suppose that you obtain a loan of P dollars (the principal value) to purchase a home. The home mortgage has an annual fixed interest rate of R% and a repayment period of M months.
1. (6 points) Let x(k) represent the remaining loan amount after the kth month, and let b represent the monthly payment. Show that x satisfies the first-order difference equation
x(k + 1) = ax(k) - b,
and specify the value of a in terms of R (note that R% = R/100).
2. (8 points) Using the initial condition x(0) = P and the Z-transform method, solve for x(k) for k ? 0. Express your answer in terms of P, a, and b.
3. (3 points) Using x(M) = 0 and your result from part 2, solve for the monthly payment b.
4. (3 points) Compute the monthly payment for a 3.6% fixed rate (a = 1.003), 30 year loan with principal amount of $225,000 (you may verify your answer using an on-line mortgage calculator).
5. (2 points) Determine the total amount of interest paid for the loan in part 3.

Problem 3: (22 points)
Suppose that you obtain a loan of P dollars (the principal value) to purchase a home. The home mortgage has an annual fixed interest rate of R% and a repayment period of M months.
1. (6 points) Let x(k) represent the remaining loan amount after the kth month, and let b represent the monthly payment. Show that x satisfies the first-order difference equation
x(k + 1) = ax(k) - b,
and specify the value of a in terms of R (note that R% = R/100).
2. (8 points) Using the initial condition x(0) = P and the Z-transform method, solve for x(k) for k ? 0. Express your answer in terms of P, a, and b.
3. (3 points) Using x(M) = 0 and your result from part 2, solve for the monthly payment b.
4. (3 points) Compute the monthly payment for a 3.6% fixed rate (a = 1.003), 30 year loan with principal amount of $225,000 (you may verify your answer using an on-line mortgage calculator).
5. (2 points) Determine the total amount of interest paid for the loan in part 3.

Added by Timothy N.

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