Kelly is receiving a 30 year annuity which makes payments at the end of each year. Payments alternate between 2,000 and 3,000. In other words, the payments at the end of years 1, 3, 5, etc are 2,000 while the payments at the end of years 2, 4, 6, etc are 3,000.

Kelly invests each payment in a fund earning a 6.75% annual effective interest rate.

How much does Kelly have at the end of 30 years.

(Round your answer to two decimal places)

To solve this problem, we will calculate the future value of the alternating annuity payments at the end of 30 years, with each payment earning an annual effective interest rate of 6.75%.

### Step-by-Step Breakdown:

1. **Identify the type of annuity**: 
   - Payments alternate between 2,000 and 3,000 at the end of each year.
   - The interest rate is 6.75%, and each payment grows until the end of 30 years.

2. **Separate the two payment streams**: 
   - One stream consists of the payments of 2,000 at the end of odd-numbered years (years 1, 3, 5, etc.).
   - The other stream consists of the payments of 3,000 at the end of even-numbered years (years 2, 4, 6, etc.).

Each stream can be treated separately as a geometric series of payments growing with interest.

3. **Use the future value formula for each stream**:
   The future value of each payment is calculated using the future value of a single payment formula:
   \[
   FV = P(1 + i)^n
   \]
   where:
   - $P$ is the payment amount,
   - $i = 6.75\%$ is the annual interest rate (as a decimal),
   - $n$ is the number of years from the payment date to the end of 30 years.

4. **Calculate the future value of each payment stream**:
   Let's calculate the future value of the odd and even payments separately. For odd payments ($P = 2000$) and even payments ($P = 3000$), we need to sum the future values of all payments.

I'll now compute this.The total amount Kelly will have at the end of 30 years is **$224,317.30**.

Would you like more details on how this was calculated, or do you have any questions?

Here are 5 related questions for further exploration:
1. How would the result change if the interest rate was different, say 5% or 8%?
2. What if the annuity was only for 20 years instead of 30?
3. How does the future value of an annuity differ from the present value?
4. What happens if the payments were made at the beginning of each year instead of the end?
5. How would inflation impact the future value of Kelly's investments?

**Tip:** When dealing with alternating payments in annuities, break them into separate streams and calculate each individually to simplify the problem.

Kelly is receiving a 30-year annuity which makes payments at the end of each year. Payments alternate between 2,000 and 3,000. In other words, the payments at the end of years 1, 3, 5, etc., are 2,000, while the payments at the end of years 2, 4, 6, etc., are 3,000. Kelly invests each payment in a fund earning a 6.75% annual effective interest rate. How much does Kelly have at the end of 30 years? (Round your answer to two decimal places)

This problem explores the future value of a 30-year annuity with alternating payments of 2,000 and 3,000, invested at a 6.75% interest rate. Detailed steps are provided, showing how to calculate the future value of the investment using compound interest principles. Suitable for finance students or advanced high school learners.

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