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To solve this problem, let's break it down:

Jeremy loans $2,500 to his son, and the son repays the loan in **5 annual payments**. Each payment consists of **interest on the outstanding balance** and an equal amount of principal repayment. Jeremy then deposits the payments into a fund that accumulates at the same interest rate.

Given that the final repayment is **$517.50**, we are asked to calculate how much the fund has accumulated to by the end of 5 years.

### Step-by-Step Breakdown:

1. **Loan Principal and Equal Principal Repayment:**
   - Total loan = $2,500
   - The son makes 5 annual payments, meaning the total principal repayment over 5 years is divided equally:
   \[
   \text{Annual principal repayment} = \frac{2500}{5} = 500
   \]
   Each year, the son repays $500 of the principal.

2. **Interest on Outstanding Balance:**
   The interest portion of each payment is calculated based on the outstanding balance. Since the final payment is given as $517.50, the interest rate can be derived as follows:
   
   - The outstanding principal at the start of year 5 is $500 (because $500 is repaid each year).
   - Hence, the interest on this amount is $517.50 - $500 = $17.50.
   
   The interest rate $ r $ can be found using:
   \[
   r = \frac{17.50}{500} = 0.035 \quad \text{or} \quad 3.5\%
   \]

3. **Yearly Payments:**
   - Year 1: The outstanding principal is $2,500, so the interest is $ 2500 \times 0.035 = 87.50 $. The total payment in year 1 is $ 500 + 87.50 = 587.50 $.
   - Year 2: The outstanding principal is $2,000, so the interest is $ 2000 \times 0.035 = 70 $. The total payment in year 2 is $ 500 + 70 = 570 $.
   - Year 3: The outstanding principal is $1,500, so the interest is $ 1500 \times 0.035 = 52.50 $. The total payment in year 3 is $ 500 + 52.50 = 552.50 $.
   - Year 4: The outstanding principal is $1,000, so the interest is $ 1000 \times 0.035 = 35 $. The total payment in year 4 is $ 500 + 35 = 535 $.
   - Year 5: The outstanding principal is $500, and as given, the final payment is $517.50.

4. **Total Fund Accumulation:**
   Each annual payment is deposited into a fund that grows at 3.5% interest. To find how much the fund has accumulated by the end of 5 years, we need to account for the compound interest on each year's deposit.

   The formula for compound interest is:
   \[
   A = P(1 + r)^t
   \]
   where:
   - $ A $ is the amount after $ t $ years,
   - $ P $ is the principal (deposit),
   - $ r $ is the annual interest rate (3.5% or 0.035),
   - $ t $ is the number of years until the end of year 5.

   Using this formula, the accumulated amount for each deposit is:
   - Year 1: $ 587.50(1.035)^4 $
   - Year 2: $ 570(1.035)^3 $
   - Year 3: $ 552.50(1.035)^2 $
   - Year 4: $ 535(1.035)^1 $
   - Year 5: $ 517.50(1.035)^0 = 517.50 $

Let's calculate the total accumulation.The total amount accumulated in the fund by the end of 5 years is approximately **$2,969.22**.

Thus, the correct option is **E: 2,969**.

Would you like more details or have any questions? Here are 5 related questions for practice:

1. How would the total accumulation change if the interest rate was 5% instead of 3.5%?
2. What would be the total repayment if the son repaid the entire loan in 3 years?
3. How much interest would be earned in the fund from the final year’s deposit alone?
4. How is the total payment amount calculated when both principal and interest are involved?
5. What is the formula to calculate the outstanding balance after each annual repayment?

**Tip:** When dealing with loans and repayments, break the problem into principal and interest components to simplify the calculations.

Jeremy loans $2,500 to his son on the condition that he repays with 5 annual payments consisting of interest on the outstanding balance and equal amounts of principal repayments. Payments are made at the end of the year. Jeremy deposits the total payment he receives into a fund that accumulates at the same interest rate. Given that his son repaid $517.50 at the end of the 5th year, how much has the fund accumulated to by the end of the 5 years?

This problem involves calculating the accumulation of a loan repayment fund with compound interest over 5 years. Given an initial loan of $2500 with equal annual principal repayments, learn to find the total accumulation in the fund after 5 years at an interest rate of 3.5%. Suitable for Grades 10-12.

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