An account credits interest at an effective rate of 4% for years 1-3, 5% for years 4-6, and 6% for years 7-9.
Deposits of $1,000 are made into the account at the end of each year for 9 years.
Calculate the accumulated value of the deposits at the end of 9 years.

To calculate the accumulated value of the deposits at the end of 9 years, we'll break down the problem based on the different interest rates applied during different time periods. We have deposits of $1,000 made at the end of each year for 9 years, and the interest rates vary:

- 4% for years 1-3,
- 5% for years 4-6,
- 6% for years 7-9.

The formula for the accumulated value $ A $ of a deposit $ P $ at time $ t $, at an effective interest rate $ i $, is given by:
\[
A = P(1 + i)^t
\]
However, in this case, the deposits are made annually, so the deposit made at the end of year 1 will accumulate interest for 8 years, the deposit made at the end of year 2 for 7 years, and so on.

### Step-by-step calculation:

1. **Interest rates for different periods:**
   - For deposits during years 1-3: 4% effective rate.
   - For deposits during years 4-6: 5% effective rate.
   - For deposits during years 7-9: 6% effective rate.

2. **Break down the accumulation for each deposit:**

- **Deposit made at the end of year 1** will accumulate interest for 8 years:
  \[
  1000 \times (1.04)^3 \times (1.05)^3 \times (1.06)^3
  \]
- **Deposit made at the end of year 2** will accumulate interest for 7 years:
  \[
  1000 \times (1.04)^2 \times (1.05)^3 \times (1.06)^3
  \]
- **Deposit made at the end of year 3** will accumulate interest for 6 years:
  \[
  1000 \times (1.04)^1 \times (1.05)^3 \times (1.06)^3
  \]
- **Deposit made at the end of year 4** will accumulate interest for 5 years:
  \[
  1000 \times (1.05)^2 \times (1.06)^3
  \]
- **Deposit made at the end of year 5** will accumulate interest for 4 years:
  \[
  1000 \times (1.05)^1 \times (1.06)^3
  \]
- **Deposit made at the end of year 6** will accumulate interest for 3 years:
  \[
  1000 \times (1.06)^3
  \]
- **Deposit made at the end of year 7** will accumulate interest for 2 years:
  \[
  1000 \times (1.06)^2
  \]
- **Deposit made at the end of year 8** will accumulate interest for 1 year:
  \[
  1000 \times (1.06)^1
  \]
- **Deposit made at the end of year 9** will not accumulate interest since it's made at the end of year 9.

### Total accumulated value:
To get the total value at the end of year 9, sum up the accumulated value of each of the deposits.

Let me calculate this for you.The total accumulated value of the deposits at the end of 9 years is approximately **$11,414.34**.

Would you like further details on any specific part of the calculation? Here are some related questions to expand your understanding:

1. How does effective interest differ from nominal interest rates?
2. What is the general formula for future value with varying interest rates?
3. How would the accumulated value change if the interest rates were compounded monthly instead of annually?
4. How does changing the timing of deposits affect the final accumulated value?
5. What happens to the accumulated value if you miss one or more annual deposits?

**Tip:** When dealing with varying interest rates, breaking the problem into different periods and applying the appropriate rate for each helps keep the calculations clear and accurate.

An account credits interest at an effective rate of 4% for years 1-3, 5% for years 4-6, and 6% for years 7-9. Deposits of $1,000 are made into the account at the end of each year for 9 years. Calculate the accumulated value of the deposits at the end of 9 years.

Learn how to calculate the accumulated value of $1,000 annual deposits over 9 years with changing interest rates: 4% for years 1-3, 5% for years 4-6, and 6% for years 7-9. This example uses compound interest formulas and effective rates to provide a detailed solution. Suitable for Grades 10-12.

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