Use PMT \( =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) to determine the regular payment amount, rounded to the nearest cent. The cost of a home is financed with a \( \$ 200,000 \) 20-year fixed-rate mortgage at \( 4.5 \% \). a. Find the monthly payments and the total interest for the loan. b. Prepare a loan amortization schedule for the first three months of the mortgage. a. The monthly payment is \( \$ \square \). (Do not round until the final answer. Then round to the nearest cent as needed.)

To find the monthly payment, we can use the provided PMT formula. Parameters: - \( P = 200,000 \) (loan principal) - \( r = 0.045 \) (annual interest rate) - \( n = 12 \) (number of payments per year) - \( t = 20 \) (duration of the loan in years) First, we need to compute \( \frac{r}{n} \): \[ \frac{r}{n} = \frac{0.045}{12} = 0.00375 \] Next, we calculate \( n \times t \): \[ n \times t = 12 \times 20 = 240 \] Now, we can substitute these values into the PMT formula: \[ PMT = \frac{200,000 \times 0.00375}{1 - (1 + 0.00375)^{-240}} \] \[ PMT = \frac{750}{1 - (1.00375)^{-240}} \] Calculating \( (1.00375)^{-240} \): \[ (1.00375)^{-240} \approx 0.39679 \] So, \[ 1 - (1.00375)^{-240} = 1 - 0.39679 = 0.60321 \] Now substitute back to find PMT: \[ PMT = \frac{750}{0.60321} \approx 1244.57 \] Thus, the monthly payment is approximately \( \text{PMT} \approx \$1244.57 \). To calculate the total interest over the life of the loan, we first determine the total payment made over 20 years: \[ \text{Total Payment} = PMT \times n \times t = 1244.57 \times 240 \approx 298,700.80 \] Now subtract the principal to find the total interest: \[ \text{Total Interest} = \text{Total Payment} - P = 298,700.80 - 200,000 = 98,700.80 \] The monthly payment is \( \$1244.57 \), and the total interest for the loan is \( \$98,700.80 \). For the loan amortization schedule for the first three months: **Month 1:** - Interest Payment: \( 200,000 \times 0.00375 = 750 \) - Principal Payment: \( 1244.57 - 750 = 494.57 \) - Remaining Balance: \( 200,000 - 494.57 = 199,505.43 \) **Month 2:** - Interest Payment: \( 199,505.43 \times 0.00375 \approx 748.90 \) - Principal Payment: \( 1244.57 - 748.90 \approx 495.67 \) - Remaining Balance: \( 199,505.43 - 495.67 = 199,009.76 \) **Month 3:** - Interest Payment: \( 199,009.76 \times 0.00375 \approx 746.29 \) - Principal Payment: \( 1244.57 - 746.29 \approx 498.28 \) - Remaining Balance: \( 199,009.76 - 498.28 = 198,511.48 \) Thus, the first three months of the amortization schedule is: - **Month 1:** Interest: \$750, Principal: \$494.57, Balance: \$199,505.43 - **Month 2:** Interest: \$748.90, Principal: \$495.67, Balance: \$199,009.76 - **Month 3:** Interest: \$746.29, Principal: \$498.28, Balance: \$198,511.48