**Transcribed Image Text: **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 \( n \)th payment to principal as \( -P^{\prime}(n) \). At that point in the course, we didn't have the tools to compute geometric sums. Now that we do, we can re-investigate our mortgage model. Let \( \$ P \) be the initial loan, and let \( r \) 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. \[ \begin{aligned} & r P+k_{1} \\ = & r\left(P-k_{1}\right)+k_{2} \\ = & r\left(P-k_{1}-k_{2}\right)+k_{3} \\ = & \cdots \\ = & r\left(P-k_{1}-k_{2}-\cdots-k_{n-1}\right)+k_{n} \\ = & r\left(P-k_{1}-k_{2}-\cdots-k_{n-1}-k_{n}\right)+k_{n+1} \\ = & \cdots \end{aligned} \] Your answer should include the interpretation of the two terms added in each line, and an explanation of why the lines are all equal. (b) Find \( k_{n+1} \) in terms of \( k_{n} \) and \( r \), using the system of equations in (a). (Assume \( 1 \leq n+1 \leq N \).) (c) Find \( k_{n} \) in terms of \( r \) and \( k_{1} \), using your answer from (b). (Assume \( n \leq N \).) (d) Interpret \( \sum_{i=1}^{N} k_{i} \) in terms of the model. (e) Find the total monthly payment amount (interest plus principal) in terms of \( P, N \), and \( r \). Hint: in small class 7, you saw a formula for geometric sums: \( \sum_{n=0}^{N} r^{n}=\frac{1-r^{N+1}}{1-r} \). (f) In class, we set \( r=\frac{1 / 4}{100}, P=750,000 \), and \( N=300 \). i. With these values of \( r, P \), and \( N \), what is the estimated monthly payment? Use a calculator to evaluate your final answer, rounding to the nearest cent.

**More** **Transcribed Image Text: **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 \( n \)th payment to principal as \( -P^{\prime}(n) \). At that point in the course, we didn't have the tools to compute geometric sums. Now that we do, we can re-investigate our mortgage model. Let \( \$ P \) be the initial loan, and let \( r \) 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. \[ \begin{aligned} & r P+k_{1} \\ = & r\left(P-k_{1}\right)+k_{2} \\ = & r\left(P-k_{1}-k_{2}\right)+k_{3} \\ = & \cdots \\ = & r\left(P-k_{1}-k_{2}-\cdots-k_{n-1}\right)+k_{n} \\ = & r\left(P-k_{1}-k_{2}-\cdots-k_{n-1}-k_{n}\right)+k_{n+1} \\ = & \cdots \end{aligned} \] Your answer should include the interpretation of the two terms added in each line, and an explanation of why the lines are all equal. (b) Find \( k_{n+1} \) in terms of \( k_{n} \) and \( r \), using the system of equations in (a). (Assume \( 1 \leq n+1 \leq N \).) (c) Find \( k_{n} \) in terms of \( r \) and \( k_{1} \), using your answer from (b). (Assume \( n \leq N \).) (d) Interpret \( \sum_{i=1}^{N} k_{i} \) in terms of the model. (e) Find the total monthly payment amount (interest plus principal) in terms of \( P, N \), and \( r \). Hint: in small class 7, you saw a formula for geometric sums: \( \sum_{n=0}^{N} r^{n}=\frac{1-r^{N+1}}{1-r} \). (f) In class, we set \( r=\frac{1 / 4}{100}, P=750,000 \), and \( N=300 \). i. With these values of \( r, P \), and \( N \), what is the estimated monthly payment? Use a calculator to evaluate your final answer, rounding to the nearest cent.