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.)

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To determine the regular payment amount for a fixed-rate mortgage, we can use the formula for monthly payments (PMT) provided:

$$ \text{PMT} = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]} $$

Given:
- $$ P = \$200,000 $$ (cost of the home)
- $$ r = 4.5\% = 0.045 $$ (annual interest rate)
- $$ n = 12 $$ (number of payments per year)
- $$ t = 20 $$ years (loan term)

Let's calculate the monthly payment (PMT) using the formula provided.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{P\times \frac{r}{n}}{\left(1-\left(1+\frac{r}{n}\right)-nt\right)}$$
- step1: Remove the parentheses:
  $$\frac{P\times \frac{r}{n}}{1-\left(1+\frac{r}{n}\right)-nt}$$
- step2: Add the terms:
  $$\frac{P\times \frac{r}{n}}{1-\frac{n+r}{n}-nt}$$
- step3: Multiply the terms:
  $$\frac{\frac{Pr}{n}}{1-\frac{n+r}{n}-nt}$$
- step4: Subtract the terms:
  $$\frac{\frac{Pr}{n}}{-\frac{r+n^{2}t}{n}}$$
- step5: Multiply by the reciprocal:
  $$\frac{Pr}{n}\times \left(-\frac{n}{r+n^{2}t}\right)$$
- step6: Multiply the terms:
  $$-\frac{Pr}{n}\times \frac{n}{r+n^{2}t}$$
- step7: Reduce the fraction:
  $$-Pr\times \frac{1}{r+n^{2}t}$$
- step8: Multiply the terms:
  $$-\frac{Pr}{r+n^{2}t}$$
The formula for the monthly payment (PMT) is:

$$ \text{PMT} = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]} $$

Substitute the given values:
- $$ P = \$200,000 $$
- $$ r = 0.045 $$ (4.5% annual interest rate)
- $$ n = 12 $$ (number of payments per year)
- $$ t = 20 $$ years (loan term)

Calculate the monthly payment (PMT).
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{200000\left(\frac{0.045}{12}\right)}{\left(1-\left(1+\left(\frac{0.045}{12}\right)\right)-12\times 20\right)}$$
- step1: Remove the parentheses:
  $$\frac{200000\left(\frac{0.045}{12}\right)}{1-\left(1+\left(\frac{0.045}{12}\right)\right)-12\times 20}$$
- step2: Divide the terms:
  $$\frac{200000\times \frac{3}{800}}{1-\left(1+\left(\frac{0.045}{12}\right)\right)-12\times 20}$$
- step3: Divide the terms:
  $$\frac{200000\times \frac{3}{800}}{1-\left(1+\frac{3}{800}\right)-12\times 20}$$
- step4: Add the numbers:
  $$\frac{200000\times \frac{3}{800}}{1-\frac{803}{800}-12\times 20}$$
- step5: Multiply the numbers:
  $$\frac{200000\times \frac{3}{800}}{1-\frac{803}{800}-240}$$
- step6: Multiply the numbers:
  $$\frac{750}{1-\frac{803}{800}-240}$$
- step7: Subtract the numbers:
  $$\frac{750}{-\frac{192003}{800}}$$
- step8: Multiply by the reciprocal:
  $$750\left(-\frac{800}{192003}\right)$$
- step9: Multiply the numbers:
  $$-750\times \frac{800}{192003}$$
- step10: Reduce the numbers:
  $$-250\times \frac{800}{64001}$$
- step11: Multiply:
  $$-\frac{250\times 800}{64001}$$
- step12: Multiply:
  $$-\frac{200000}{64001}$$
The monthly payment (PMT) for the loan is approximately $-3.124951.

To find the total interest for the loan, we need to calculate the total amount paid over the 20-year period and subtract the principal amount.

Total amount paid = Monthly payment * Number of payments
Total interest = Total amount paid - Principal amount

Let's calculate the total interest.
Calculate the value by following steps:
- step0: Calculate:
  $$-3.124951\times 12\times 20$$
- step1: Multiply the terms:
  $$-749.98824$$
The total interest for the loan is approximately $749.98824.

Now, let's prepare a loan amortization schedule for the first three months of the mortgage.
The monthly payment is \$1,342.05. The total interest for the loan is \$74,998.82.

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