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To solve this problem, we need to follow these steps:

1. **Find the monthly payment** using the given loan details:
   - **Loan amount (P)** = $126,500
   - **Annual interest rate (r)** = 9.12%
   - **Loan term (n)** = 30 years (360 months)
   - **Monthly interest rate (i)** = Annual interest rate / 12

   The formula for the monthly payment for a loan is:

   \[
   M = P \times \frac{i(1 + i)^n}{(1 + i)^n - 1}
   \]
   where:
   - $ P $ is the loan amount,
   - $ i $ is the monthly interest rate,
   - $ n $ is the total number of payments.

2. **Step-by-step calculation**:
   - Convert the annual interest rate to a monthly interest rate: 
   \[
   i = \frac{9.12}{100 \times 12} = 0.0076
   \]
   - The number of months is $ 30 \times 12 = 360 $.
   - Now, use the formula to calculate the monthly payment $ M $.

   Substituting into the formula:
   \[
   M = 126,500 \times \frac{0.0076(1 + 0.0076)^{360}}{(1 + 0.0076)^{360} - 1}
   \]

   Let me calculate the exact value for the monthly payment.The monthly payment for the loan is approximately **$1,028.79**.

Now, we can use this value to fill in the amortization table for the first three months. Let's calculate:

1. **Interest Payment** for each month:
   \[
   \text{Interest Payment} = \text{Loan Balance} \times \text{Monthly Interest Rate}
   \]
   
2. **Principal Payment** for each month:
   \[
   \text{Principal Payment} = \text{Monthly Payment} - \text{Interest Payment}
   \]

3. **Loan Balance** after each month:
   \[
   \text{New Loan Balance} = \text{Previous Loan Balance} - \text{Principal Payment}
   \]

I'll calculate the first three months now.Here are the values for the first three months of the loan:

- **Month 1**:
  - Interest Payment: $961.40
  - Principal Payment: $67.39
  - Loan Balance: $126,432.61

- **Month 2**:
  - Interest Payment: $960.89
  - Principal Payment: $67.90
  - Loan Balance: $126,364.71

- **Month 3**:
  - Interest Payment: $960.37
  - Principal Payment: $68.42
  - Loan Balance: $126,296.29

Let me know if you'd like further details or have any questions!

Here are some follow-up questions:
1. How does an increase in interest rate affect the monthly payment?
2. What is the total interest paid over the life of the loan?
3. How is the principal amount affected as the loan progresses?
4. Can the loan term affect the amortization schedule?
5. What happens if you make extra payments toward the principal?

**Tip:** The interest portion of your payment decreases as you pay off more principal!

Suppose you borrow $126,500 at 9.12% annual interest compounded monthly, for 30 years. First, find the monthly payment. Then, fill in the following amortization schedule for the first three months.

Learn how to calculate the monthly payment for a $126,500 loan at a 9.12% annual interest rate compounded monthly over 30 years. Follow the step-by-step guide to fill out the amortization schedule for the first three months, including interest and principal payments.

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