Using an annual effective interest rate of 9.8%, calculate the accumulated value of an annuity that pays 250 at the beginning of each month for 14 years.

(Round your answer to the nearest two decimal places)

To calculate the accumulated value of an annuity that pays 250 at the beginning of each month for 14 years with an annual effective interest rate of 9.8%, we need to use the formula for the future value of an annuity due (because payments are made at the beginning of each period).

### Future Value of Annuity Due Formula:
\[
FV = P \times \left( \frac{(1 + i)^n - 1}{i} \right) \times (1 + i)
\]

Where:
- $ P = 250 $ (the monthly payment),
- $ i $ is the monthly effective interest rate,
- $ n = 14 \times 12 = 168 $ (the total number of payments),
- $ FV $ is the accumulated value of the annuity.

### Steps:
1. First, we need to convert the annual effective interest rate of 9.8% to a monthly effective interest rate. Since the rate is compounded monthly, we use the formula:

\[
i = (1 + r)^{\frac{1}{12}} - 1
\]
Where $ r = 0.098 $.

2. Then, we can substitute the values into the future value formula and calculate the accumulated value.

Let's calculate this now.The accumulated value of the annuity after 14 years, with payments of 250 made at the beginning of each month and an annual effective interest rate of 9.8%, is **$87,041.33**.

Would you like further details on the calculation or have any other questions?

Here are 5 related questions to explore further:
1. How does the accumulated value change with a higher or lower interest rate?
2. What is the difference between an annuity due and an ordinary annuity?
3. How would the accumulated value change if the payments were made at the end of each month?
4. What is the impact of making extra payments during the annuity period?
5. How can the present value of this annuity be calculated?

**Tip:** In financial mathematics, understanding the difference between annuities due and ordinary annuities can significantly affect the results of accumulated and present values.

Using an annual effective interest rate of 9.8%, calculate the accumulated value of an annuity that pays 250 at the beginning of each month for 14 years. (Round your answer to the nearest two decimal places)

Discover how to calculate the accumulated value of an annuity that pays $250 at the beginning of each month for 14 years, using a 9.8% annual effective interest rate. This detailed explanation covers the Future Value of Annuity Due formula and compound interest, suitable for advanced high school and college-level students.

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