You need $25,656 at the end of 7 years, and your only investment outlet is an 12 percent long-term certificate of deposit (compounded annually). With the certificate of deposit, you make an initial investment at the beginning of the first year. What amount could you pay at the end of each year annually for 7 years to achieve this same objective?

To determine the amount you would need to pay at the end of each year for 7 years to accumulate $25,656 at a 12% annual interest rate, we can use the formula for the **Future Value of an Annuity (FVA)**. The formula for FVA is:

\[
FVA = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $ FVA $ is the future value of the annuity (which is $25,656 in this case),
- $ P $ is the annuity payment (the amount you pay at the end of each year),
- $ r $ is the annual interest rate (12% or 0.12),
- $ n $ is the number of years (7 years).

Rearranging the formula to solve for $ P $:

\[
P = \frac{FVA \times r}{(1 + r)^n - 1}
\]

Now, substituting the given values:

\[
P = \frac{25,656 \times 0.12}{(1 + 0.12)^7 - 1}
\]

Let's calculate that.The amount you would need to pay at the end of each year for 7 years to accumulate $25,656 at a 12% annual interest rate is approximately **$2,542.96**.

Would you like further details or have any questions?

Here are 5 related questions you might find useful:
1. How does the interest rate affect the annuity payment amount?
2. What happens if the interest is compounded more frequently, like monthly or quarterly?
3. How would the result change if the payments were made at the beginning of each year?
4. How can you calculate the total amount of payments made over the 7 years?
5. What if the target amount changes to $30,000 with the same interest rate and duration?

**Tip:** Always verify whether interest is compounded annually, monthly, or quarterly, as this significantly impacts investment growth.

Learn how to calculate the annual annuity payments required to accumulate $25,656 in 7 years at a 12% interest rate, compounded annually. This finance problem uses the future value of annuity formula and applies compound interest principles, suitable for college-level finance and mathematics students.

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