a.a.

Use the appropriate formula to find the value of the annuity.

b.b.

Find the interest.

**Periodic Deposit**

**Rate**

**Time**

​$20002000

at the end of every three months

7.25 %7.25%

compounded quarterly

55

years

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Part 1

a.a.

The value of the annuity is

​$enter your response here.

​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)

Part 2

b.b.

The interest is $ nothing .$enter your response here.

​(Use the answer from part​ (a) to find this answer. Round to the nearest dollar as​ needed.)

To solve this problem, we'll calculate the future value of the annuity and then the total interest earned. Here's the step-by-step process:

### Part 1: Calculate the Value of the Annuity

Given:
- **Periodic Deposit (PMT)**: $2000
- **Interest Rate per Period (i)**: $ \text{Rate per year} = 7.25\% $, compounded quarterly
- **Time (n)**: 5 years

First, determine the interest rate per period and the number of periods:
- Since it's compounded quarterly, each period is 3 months, or 4 periods per year.
- The interest rate per period: $ i = \frac{7.25\%}{4} = \frac{0.0725}{4} = 0.018125 $ per period.
- The number of periods: $ n = 5 \times 4 = 20 $ periods.

The formula for the future value of an ordinary annuity (deposits made at the end of each period) is:

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

Substitute the values into the formula:

\[
FV = 2000 \times \frac{(1 + 0.018125)^{20} - 1}{0.018125}
\]

\[
FV = 2000 \times \frac{(1.018125)^{20} - 1}{0.018125}
\]

Now, calculate the exact value.

### Part 2: Calculate the Interest Earned

The total interest earned is the difference between the future value of the annuity and the total amount deposited:

\[
\text{Interest} = FV - \text{Total Deposits}
\]

The total deposits are:

\[
\text{Total Deposits} = PMT \times n = 2000 \times 20 = 40000
\]

Thus,

\[
\text{Interest} = FV - 40000
\]

Let's compute the future value first, then the interest.

### Calculations
Now, I will perform these calculations for you.### Results

**Part 1: The value of the annuity is** \$47,698 (rounded to the nearest dollar).

**Part 2: The interest earned is** \$7,698 (rounded to the nearest dollar).

If you have any questions or need more details, feel free to ask.

Here are 8 related questions that might interest you:
1. How does the future value change if the interest rate increases to 8%?
2. What if the deposits are made at the beginning of each period instead of the end?
3. How does the future value change if the time is extended to 10 years?
4. What happens to the interest earned if the periodic deposit is increased to $3000?
5. How do monthly compounding and quarterly compounding compare for this scenario?
6. How much would you need to deposit periodically to reach a future value of $100,000 in 5 years?
7. What is the effect of decreasing the interest rate to 5%?
8. How does inflation affect the real value of the future amount?

**Tip:** For larger deposits or longer periods, even small changes in the interest rate can have a significant impact on the future value of the annuity.

Learn how to calculate the future value of an annuity with quarterly compounding. This problem involves a periodic deposit of $2000 every three months, compounded quarterly at an annual interest rate of 7.25% over a period of 5 years. Discover the exact steps and formulas used in finance to solve such annuity problems.

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