To find the total value of the annuity after 3 years with annual payments of [tex]$5996, an interest rate of 2% compounded annually, we can use the future value of an annuity formula:
\[
A = M \left[ \frac{(1 + \frac{r}{n})^{nt} - 1}{\frac{r}{n}} \right]
\]
Where:
- \(A\) is the future value of the annuity,
- \(M\) is the yearly payment,
- \(r\) is the annual interest rate,
- \(n\) is the number of times the interest is compounded per year,
- \(t\) is the number of years.
Given:
- \(M = 5996\),
- \(r = 0.02\),
- \(n = 1\) (compounded annually),
- \(t = 3\),
We can plug these values into the formula to calculate the future value of the annuity.
### Step-by-Step Solution
1. First, substitute the given values into the formula:
\[
A = 5996 \left[ \frac{(1 + \frac{0.02}{1})^{1 \cdot 3} - 1}{\frac{0.02}{1}} \right]
\]
2. Compute the inside of the parenthesis:
\[
1 + \frac{0.02}{1} = 1 + 0.02 = 1.02
\]
3. Raise this value to the power of \(nt\):
\[
(1.02)^{1 \cdot 3} = (1.02)^3 = 1.061208
\]
4. Subtract 1 from this value:
\[
1.061208 - 1 = 0.061208
\]
5. Divide by \(\frac{0.02}{1}\):
\[
\frac{0.061208}{0.02} = 3.0604
\]
6. Multiply this result by the yearly payment \(M\):
\[
5996 \times 3.0604 = 18350.1568
\]
7. Round the final answer to the nearest cent:
\[
18350.16
\]
So, the total value of the annuity in 3 years is \( \$[/tex] 18350.16 \).
