Jeremy takes out a 30-year mortgage of 250000 dollars at an annual interest rate of 6.5 percent compounded monthly, with the first payment due in one month. How much does he owe on the loan immediately after the 87th payment?

Jeremy's mortgage is a standard amortizing loan, where he makes equal payments each month, and each payment is used to cover both interest and principal. We need to calculate how much he still owes after making the 87th payment.

### Step 1: Loan Parameters
- Loan Amount (Principal), $ P = 250,000 $ dollars.
- Annual interest rate, $ r = 6.5\% $ (or $ 0.065 $ as a decimal).
- Monthly interest rate, $ i = \frac{0.065}{12} = 0.00541667 $.
- Loan term, $ N = 30 $ years $ = 30 \times 12 = 360 $ months.

### Step 2: Monthly Payment Calculation
We first calculate the fixed monthly payment using the mortgage payment formula:

\[
M = P \cdot \frac{i(1 + i)^N}{(1 + i)^N - 1}
\]

Substituting the known values:

\[
M = 250,000 \cdot \frac{0.00541667(1 + 0.00541667)^{360}}{(1 + 0.00541667)^{360} - 1}
\]

Let’s calculate $ M $.

### Step 3: Remaining Loan Balance After 87 Payments
The remaining balance after $ t $ payments is calculated using the formula:

\[
B_t = P \cdot \frac{(1 + i)^N - (1 + i)^t}{(1 + i)^N - 1}
\]

Where:
- $ t = 87 $ is the number of payments made.
- $ N = 360 $ is the total number of payments.

We'll calculate both the monthly payment and the remaining balance using these formulas.

Let me compute these values.### Results:
- **Monthly payment**: \$1,580.17
- **Remaining balance after 87 payments**: \$224,967.47

Thus, Jeremy will still owe **\$224,967.47** after making the 87th payment.

Would you like a deeper breakdown or have further questions?

Here are five related questions:
1. How is the monthly interest rate derived from the annual interest rate?
2. What happens to the loan balance if the interest rate decreases by 1%?
3. How can extra payments reduce the remaining loan balance?
4. How is the amortization schedule related to the principal and interest portions of each payment?
5. What would be the effect on the loan term if the payments are increased by 10%?

**Tip**: Making extra payments directly towards the principal can significantly reduce both the loan term and the total interest paid.

Calculate the remaining loan balance after making 87 payments on a 30-year mortgage of $250,000 at 6.5% annual interest compounded monthly. This solution uses key concepts of amortization, including the mortgage payment and balance formulas, providing an in-depth analysis of loan repayment. Suitable for advanced high school or undergraduate students.

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