A loan is being paid off by payments of 1,000, 2,000, .., 10,000 at the end of years 1, 2, ..., 10

The effective annual interest rate is 18%.

Determine the amount of interest in the 7th payment.

We are given that a loan is being repaid with increasing payments over 10 years, with the effective annual interest rate at 18%. To find the amount of interest in the 7th payment, we need to break down the problem into clear steps.

### Step 1: Calculate the total loan amount
The payments made over 10 years form an arithmetic series:
- The first payment is 1,000.
- The second payment is 2,000, and so on up to the 10th payment of 10,000.

The sum of this arithmetic series can be computed using the formula:

\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]
where:
- \( n = 10 \) (number of terms),
- \( a_1 = 1,000 \) (first payment),
- \( a_{10} = 10,000 \) (last payment).

\[
S_{10} = \frac{10}{2} \cdot (1,000 + 10,000) = 5 \cdot 11,000 = 55,000
\]

Thus, the total amount paid over the 10 years is 55,000. This represents the total repayment, including principal and interest.

### Step 2: Compute the loan balance before the 7th payment
We now need to calculate how much of the loan remains just before the 7th payment. Since each payment pays off both interest and principal, we can calculate this by discounting future payments back to the 7th payment using the effective interest rate of 18%.

The balance just before the 7th payment is the present value of the remaining payments (from years 7 to 10). The remaining payments are \( 7,000 \), \( 8,000 \), \( 9,000 \), and \( 10,000 \).

The present value of these payments at time 6 (just before the 7th payment) is calculated using the formula for the present value of future payments:

\[
PV = \sum \frac{P_k}{(1 + i)^k}
\]

Where:
- \( P_k \) is the payment in year \( k \),
- \( i = 0.18 \) is the interest rate,
- \( k \) is the number of years from time 6.

The present value of the payments is:

\[
PV = \frac{7,000}{(1.18)^1} + \frac{8,000}{(1.18)^2} + \frac{9,000}{(1.18)^3} + \frac{10,000}{(1.18)^4}
\]

We can compute this step by step:

\[
PV = \frac{7,000}{1.18} + \frac{8,000}{1.18^2} + \frac{9,000}{1.18^3} + \frac{10,000}{1.18^4}
\]

\[
PV \approx 5,932.20 + 5,743.34 + 5,471.90 + 5,116.69
\]

\[
PV \approx 22,264.13
\]

Thus, the balance before the 7th payment is approximately 22,264.13.

### Step 3: Calculate the interest portion of the 7th payment
The interest portion of the 7th payment is calculated by multiplying the loan balance before the 7th payment by the interest rate:

\[
\text{Interest} = 22,264.13 \times 0.18 = 4,007.54
\]

### Final Answer:
The interest portion of the 7th payment is approximately **4,007.54**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How is the present value of a series of payments computed?
2. How does an effective interest rate differ from a nominal interest rate?
3. Can you find the total interest paid over the entire loan period?
4. How would the interest calculation change if the payments were constant?
5. What is the difference between simple interest and compound interest?

**Tip:** When calculating loan balances, always remember that the interest portion decreases as the principal is paid down, while the principal portion increases over time.

A loan is being paid off by payments of 1,000, 2,000, ..., 10,000 at the end of years 1, 2, ..., 10. The effective annual interest rate is 18%. Determine the amount of interest in the 7th payment.

Learn how to calculate the interest in the 7th payment of a loan repaid with increasing payments over 10 years at an 18% effective annual interest rate. This example involves key concepts in loan repayment, interest calculation, and present value formulas.

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