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To solve this annuity problem, let's go through the formula step-by-step. The formula given for the future value of an ordinary annuity is:

\[
A = m \left( \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}} \right)
\]

where:
- $ m $ is the monthly payment,
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of compounding periods per year,
- $ t $ is the number of years.

Given:
- Monthly payment, $ m = 50 $ dollars,
- Annual interest rate, $ r = 8\% = 0.08 $,
- Number of compounding periods per year, $ n = 12 $ (for monthly payments),
- Time period, $ t = 3 $ years.

### Step 1: Substitute the values into the formula

\[
A = 50 \left( \frac{\left(1 + \frac{0.08}{12}\right)^{12 \times 3} - 1}{\frac{0.08}{12}} \right)
\]

### Step 2: Calculate the terms inside the formula

1. **Calculate** $ \frac{r}{n} $:
   \[
   \frac{r}{n} = \frac{0.08}{12} \approx 0.0066667
   \]

2. **Calculate** $ \left(1 + \frac{r}{n}\right) $:
   \[
   1 + 0.0066667 \approx 1.0066667
   \]

3. **Calculate** $ (1.0066667)^{12 \times 3} $:
   \[
   (1.0066667)^{36} \approx 1.28368
   \]

4. **Subtract 1**:
   \[
   1.28368 - 1 = 0.28368
   \]

5. **Divide by $ \frac{r}{n} $**:
   \[
   \frac{0.28368}{0.0066667} \approx 42.552
   \]

### Step 3: Multiply by $ m $ to find $ A $

\[
A = 50 \times 42.552 \approx 2127.60
\]

### Final Answer:
\[
A \approx 2127.60
\]

So, the future value of this annuity is approximately **$2127.60**.

Would you like further details on any part of the solution, or do you have any questions?

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Here are five additional questions related to this topic:

1. What would the future value be if the interest rate was 5% instead of 8%?
2. How would the annuity formula change if payments were made quarterly instead of monthly?
3. What would happen to the future value if the payment amount increased to $100?
4. How does the compounding frequency impact the overall future value in annuity calculations?
5. If the annuity continued for 5 years instead of 3, how much would the future value increase?

**Tip:** For accurate results, break down complex formulas into steps when dealing with financial calculations involving powers and divisions.

Use a calculator to evaluate an ordinary annuity formula. For m, r, and t (respectively). Assume monthly payments. Round your answer to the nearest cent. Given: $50; 8%; 3 years.

Calculate the future value of an ordinary annuity with monthly payments of $50 over 3 years at an 8% annual interest rate. This problem uses the ordinary annuity formula, illustrating compound interest calculations with monthly compounding.

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