Find the value of the ordinary annuity at the end of the indicated time period. The payment​ R, frequency of deposits m​ (which is the same as the frequency of​ compounding), annual interest rate​ r, and time period t are given.
amount ​$500​; quarterly​; 6.1​%; five years. do not round until final answer


To find the value of an ordinary annuity at the end of the indicated time period, we use the formula:

\[
A = R \times \frac{(1 + \frac{r}{m})^{mt} - 1}{\frac{r}{m}}
\]

where:
- $ A $ is the future value of the annuity,
- $ R $ is the regular payment,
- $ r $ is the annual interest rate (in decimal form),
- $ m $ is the number of compounding periods per year,
- $ t $ is the number of years.

### Given:
- $ R = 500 $ (payment),
- $ r = 6.1\% = 0.061 $ (annual interest rate),
- $ m = 4 $ (quarterly deposits/compounding),
- $ t = 5 $ years.

### Step-by-step solution:

1. First, calculate the interest per compounding period:  
   \[
   \frac{r}{m} = \frac{0.061}{4} = 0.01525
   \]

2. Calculate the total number of compounding periods:  
   \[
   mt = 4 \times 5 = 20
   \]

3. Substitute these values into the annuity formula:

\[
A = 500 \times \frac{(1 + 0.01525)^{20} - 1}{0.01525}
\]

4. Calculate $ (1 + 0.01525)^{20} $:

\[
(1 + 0.01525)^{20} = (1.01525)^{20} \approx 1.346855
\]

5. Subtract 1 from the result:

\[
1.346855 - 1 = 0.346855
\]

6. Now divide by $ 0.01525 $:

\[
\frac{0.346855}{0.01525} \approx 22.742
\]

7. Multiply by $ R = 500 $:

\[
A = 500 \times 22.742 \approx 11,371.06
\]

### Final Answer:
The future value of the annuity is approximately **$11,371.06**.

Would you like more details or have any questions? Here are 5 related questions to expand this topic:

1. What if the deposits were made monthly instead of quarterly?
2. How does the annuity value change if the interest rate was increased to 8%?
3. How do we compute the value of a *present* annuity?
4. What happens to the future value if the time period is extended to 10 years?
5. How would you calculate the annuity if compounding was done continuously?

**Tip:** When working with annuities, always ensure the interest rate is expressed in decimal form before calculating.

Find the value of the ordinary annuity at the end of the indicated time period. The payment​ R, frequency of deposits m​ (which is the same as the frequency of​ compounding), annual interest rate​ r, and time period t are given. Amount ​$500​; quarterly​; 6.1​%; five years. Do not round until final answer.

Learn how to calculate the future value of an ordinary annuity with $500 quarterly payments, an interest rate of 6.1%, and a 5-year time period. Follow a detailed step-by-step solution using the annuity formula. Ideal for students studying finance or advanced high school math.

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