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Part 1
a. Use the appropriate formula to find the value of the annuity.
b. Find the interest.
Periodic Deposit
Rate
Time
​$120 at the end of every six months
4.5 % compounded semiannually
10 years
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Part 1
a. The value of the annuity is ​$
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)

We are asked to find the value of an annuity with the following details:

- **Periodic deposit (PMT)**: $120
- **Interest rate (annual)**: 4.5% compounded semiannually
- **Time (years)**: 10 years

### Part 1a: Formula to Find the Value of the Annuity

Since the interest is compounded semiannually and payments are made at the end of each period, we will use the formula for the future value of an ordinary annuity:

\[
FV = PMT \times \frac{\left( (1 + r/n)^{nt} - 1 \right)}{r/n}
\]

Where:
- $ FV $ is the future value (the value of the annuity we need to find),
- $ PMT $ is the periodic deposit ($120),
- $ r $ is the annual interest rate (0.045),
- $ n $ is the number of compounding periods per year (2 for semiannual),
- $ t $ is the time in years (10 years).

Let's plug in the values:
- $ r = 0.045 $,
- $ n = 2 $,
- $ t = 10 $,
- $ PMT = 120 $.

\[
FV = 120 \times \frac{\left( (1 + 0.045/2)^{2 \times 10} - 1 \right)}{0.045/2}
\]

\[
FV = 120 \times \frac{\left( (1 + 0.0225)^{20} - 1 \right)}{0.0225}
\]

Now, calculate the value inside the brackets.

\[
(1 + 0.0225)^{20} = 1.0225^{20} \approx 1.563717
\]

So,

\[
FV = 120 \times \frac{(1.563717 - 1)}{0.0225}
\]

\[
FV = 120 \times \frac{0.563717}{0.0225}
\]

\[
FV = 120 \times 25.0541 \approx 3006.49
\]

Rounding to the nearest dollar:

\[
FV \approx 3006
\]

So, the value of the annuity is **$3,006**.

---

### Part 1b: Find the Interest

To find the total interest earned, subtract the total deposits made from the future value of the annuity. The total deposits are:

\[
\text{Total deposits} = PMT \times \text{number of periods} = 120 \times (2 \times 10) = 120 \times 20 = 2400
\]

So, the interest earned is:

\[
\text{Interest} = FV - \text{Total deposits} = 3006 - 2400 = 606
\]

Thus, the interest earned is **$606**.

---

Would you like more details or have any other questions?

Here are some related questions:
1. What would the annuity's value be if compounded quarterly?
2. How would the future value change if the interest rate was 5%?
3. What if the periodic deposit was $150 instead of $120?
4. How would the result change if the annuity was compounded annually instead?
5. Can you calculate the total amount deposited over a 15-year period?

**Tip:** Always confirm whether an annuity is ordinary (payments at the end of periods) or due (payments at the beginning), as this affects the formula.

Part 1: a. Use the appropriate formula to find the value of the annuity. b. Find the interest. Periodic Deposit: $120 at the end of every six months, Rate: 4.5% compounded semiannually, Time: 10 years.

Learn how to calculate the future value of an annuity with $120 deposits, compounded semiannually at a 4.5% interest rate over 10 years. This step-by-step guide will show you the key annuity formula and interest earned, ideal for students in Grades 10-12.

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