a. Use the appropriate formula to find the value of the annuity. b. Find the interest. Data is as follows. Perodic deposit: $9000 at the end of each year, rate: 3.5% compounded annually, time: 20 years. The value of the annuity is $254517. Determine the interest in $

### Part a: Finding the value of the annuity

The value of an ordinary annuity (where deposits are made at the end of each period) is given by the formula:

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

where:
- $A$ = the future value of the annuity (value of the annuity),
- $P$ = periodic deposit (in this case, $9000),
- $r$ = interest rate per period (in decimal form, here $3.5\% = 0.035$),
- $t$ = number of periods (20 years).

Using the data### Part a: Finding the value of the annuity

The value of an ordinary annuity (where deposits are made at the end of each period) is given by the formula:

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

where:
- $A$ = the future value of the annuity (value of the annuity),
- $P$ = periodic deposit (in this case, $9000),
- $r$ = interest rate per period (in decimal form, here $3.5\% = 0.035$),
- $t$ = number of periods (20 years).

Using the 
\[
A = 9000 \times \frac{(1 + 0.035)^{20} - 1}{0.035}
\]

Now, solving this equation will give us the future value, but it's already given that $A = 254517$. Therefore, this is confirmed as the value of the annuity.

### Part b: Finding the interest

The interest earned on an annuity is the difference between the total amount in the annuity and the total deposits made. The formula for the interest is:

\[
\text{Interest} = A - (P \times t)
\]

Where:
- $A = 254517$ (value of the annuity),
- $P = 9000$ (periodic deposit),
- $t = 20$ (number of periods).

Now, substituting these values into the formula:

\[
\text{Interest} = 254517 - (9000 \times 20)
\]
\[
\text{Interest} = 254517 - 180000
\]
\[
\text{Interest} = 74517
\]

### Final Answer:
- **Interest** earned over 20 years is **$74,517**.

Would you like a detailed explanation of the formulas used, or have any further questions?

Here are 5 related questions:
1. What would the annuity value be if the interest rate increased to 5%?
2. How does the compounding frequency affect the annuity's future value?
3. What would the total interest be if deposits were made at the beginning of each year?
4. How does reducing the time period to 15 years affect the total annuity and interest earned?
5. What is the formula for calculating the present value of an annuity?

**Tip:** Always check if deposits are made at the end or beginning of the period, as it changes the formula!

Use the appropriate formula to find the value of the annuity. Find the interest. Data is as follows. Periodic deposit: $9000 at the end of each year, rate: 3.5% compounded annually, time: 20 years. The value of the annuity is $254517. Determine the interest in $.

Calculate the annuity value and interest earned for $9000 periodic deposits at the end of each year with a 3.5% annual interest rate over 20 years. Learn how to apply the annuity formula and determine the total interest of $74,517.

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