One retailer charges $$ \$ 1028 $$ for a certain computer. A firm of tax accountants buys 7 of these computers. It makes a down payment of $$ \$ 1100 $$ and agrees to amortize the balance with monthly payments at $$ 15 \% $$ interest for 5 years. Prepare an amortization schedule showing the first four monthly payments for this loan. Write the equation that can be used to find the amount of each monthly payment. PMT $$ =\frac{\mathrm{PV} \cdot \square}{1-(1+\square)} $$ where PV $$ =\square $$ (Type integers or decimals. Simplify your answers.)

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To prepare an amortization schedule showing the first four monthly payments for the loan, we need to calculate the monthly payment amount using the given formula:

$$ \text{PMT} = \frac{\text{PV} \cdot r}{1 - (1 + r)^n} $$

where:
- PV is the present value (initial amount) of the loan,
- r is the monthly interest rate (15% per year divided by 12),
- n is the number of payments (5 years multiplied by 12 months per year).

Given:
- PV = $1100 (down payment)
- r = 15% per year / 12 months per year = 0.0125
- n = 5 years * 12 months per year = 60 months

Let's calculate the monthly payment amount using the formula provided.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1100\times 0.0125}{\left(1-\left(1+0.0125\right)^{60}\right)}$$
- step1: Remove the parentheses:
  $$\frac{1100\times 0.0125}{1-\left(1+0.0125\right)^{60}}$$
- step2: Add the numbers:
  $$\frac{1100\times 0.0125}{1-1.0125^{60}}$$
- step3: Convert the expressions:
  $$\frac{1100\times 0.0125}{1-\left(\frac{81}{80}\right)^{60}}$$
- step4: Multiply the numbers:
  $$\frac{13.75}{1-\left(\frac{81}{80}\right)^{60}}$$
- step5: Subtract the numbers:
  $$\frac{13.75}{\frac{80^{60}-81^{60}}{80^{60}}}$$
- step6: Convert the expressions:
  $$\frac{\frac{55}{4}}{\frac{80^{60}-81^{60}}{80^{60}}}$$
- step7: Multiply by the reciprocal:
  $$\frac{55}{4}\times \frac{80^{60}}{80^{60}-81^{60}}$$
- step8: Rewrite the expression:
  $$\frac{55}{4}\times \frac{16^{60}\times 5^{60}}{80^{60}-81^{60}}$$
- step9: Rewrite the expression:
  $$\frac{55}{4}\times \frac{4^{120}\times 5^{60}}{80^{60}-81^{60}}$$
- step10: Reduce the numbers:
  $$55\times \frac{4^{119}\times 5^{60}}{80^{60}-81^{60}}$$
- step11: Multiply:
  $$\frac{55\times 4^{119}\times 5^{60}}{80^{60}-81^{60}}$$
The monthly payment amount is approximately $12.42.

Now, let's prepare the amortization schedule for the first four monthly payments. The schedule will show the payment amount, interest paid, and principal paid for each month.
To find the monthly payment, use the formula: $$ \text{PMT} = \frac{\text{PV} \cdot r}{1 - (1 + r)^n} $$ Where: - PV = $1100 - r = 0.0125 (monthly interest rate) - n = 60 (total payments) Calculating this gives a monthly payment of approximately $12.42. Here's the amortization schedule for the first four payments: 1. **Month 1:** - Payment: $12.42 - Interest: $1100 * 0.0125 = $13.75 - Principal: $12.42 - $13.75 = -$1.33 (Note: Since the interest is higher than the payment, the principal decreases by $1.33) - Remaining Balance: $1100 - $1.33 = $1098.67 2. **Month 2:** - Payment: $12.42 - Interest: $1098.67 * 0.0125 = $13.73 - Principal: $12.42 - $13.73 = -$1.31 - Remaining Balance: $1098.67 - $1.31 = $1097.36 3. **Month 3:** - Payment: $12.42 - Interest: $1097.36 * 0.0125 = $13.72 - Principal: $12.42 - $13.72 = -$1.30 - Remaining Balance: $1097.36 - $1.30 = $1096.06 4. **Month 4:** - Payment: $12.42 - Interest: $1096.06 * 0.0125 = $13.70 - Principal: $12.42 - $13.70 = -$1.28 - Remaining Balance: $1096.06 - $1.28 = $1094.78

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