Find the future value of an ordinary annuity of $4,000 paid quarterly for 2 years, if the interest rate is 8%, compounded quarterly. (Round your answer to the nearest cent.)


To find the future value of an ordinary annuity, where payments are made at the end of each period, we can use the future value of an annuity formula:

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

where:
- $ P $ is the payment amount per period,
- $ r $ is the interest rate per period,
- $ n $ is the total number of periods.

### Step-by-Step Solution

1. **Identify the variables:**
   - **Quarterly Payment (P)**: $4,000
   - **Annual Interest Rate**: 8%
   - **Compounding Periods per Year**: Quarterly (4 times per year)
   - **Duration**: 2 years

2. **Calculate the rate per period and total number of periods:**
   - **Quarterly Interest Rate (r)**: Since the annual rate is 8%, the quarterly rate is:
     \[
     r = \frac{8\%}{4} = 2\% = 0.02
     \]
   - **Total Number of Periods (n)**: Since payments are made quarterly for 2 years:
     \[
     n = 2 \times 4 = 8 \text{ quarters}
     \]

3. **Plug these values into the formula:**
   \[
   FV = 4000 \times \frac{(1 + 0.02)^8 - 1}{0.02}
   \]

4. **Calculate the result:**
   - First, compute $ (1 + 0.02)^8 $:
     \[
     (1 + 0.02)^8 = 1.171659
     \]
   - Next, subtract 1:
     \[
     1.171659 - 1 = 0.171659
     \]
   - Then, divide by the interest rate per period (0.02):
     \[
     \frac{0.171659}{0.02} = 8.58295
     \]
   - Finally, multiply by the payment amount:
     \[
     FV = 4000 \times 8.58295 = 34331.80
     \]

### Answer
The future value of the annuity is approximately **$34,331.80**.

Would you like a deeper explanation, or do you have any questions?

Here are some related questions to expand your understanding:
1. How would the future value change if payments were made monthly instead of quarterly?
2. What would the future value be if the interest rate were compounded monthly instead of quarterly?
3. How would the formula adjust if this were an annuity due instead of an ordinary annuity?
4. What impact does increasing the interest rate have on the future value of an annuity?
5. How does the future value change if the payment amount is increased by $1,000?

**Tip:** When working with annuities, always double-check if it's an ordinary annuity or an annuity due, as this affects the timing of payments and the resulting calculations.

Find the future value of an ordinary annuity of $4,000 paid quarterly for 2 years, if the interest rate is 8%, compounded quarterly. (Round your answer to the nearest cent.)

Learn how to calculate the future value of an ordinary annuity with quarterly payments. In this example, we calculate the future value for an annuity of $4,000 paid quarterly over 2 years at an 8% annual interest rate, compounded quarterly. The future value is approximately $34,331.80.

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