To find the future value of an annuity, we use the formula for the future value of an ordinary annuity, which is:
[tex]\[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \][/tex]
where:
- [tex]\( FV \)[/tex] represents the future value of the annuity
- [tex]\( PMT \)[/tex] represents the regular payment amount
- [tex]\( r \)[/tex] represents the interest rate per period
- [tex]\( n \)[/tex] represents the number of periods
Given:
- The regular payment amount, [tex]\( PMT \)[/tex], is [tex]$2000
- The interest rate per period, \( r \), is 7% (or 0.07 when written as a decimal)
- The number of periods, \( n \), is 10 years
Let's calculate the future value step-by-step.
1. Start by finding the value of \((1 + r)^n\):
\[ (1 + 0.07)^{10} \]
2. Simplify the expression:
\[ (1.07)^{10} \]
3. Compute \((1.07)^{10}\):
\[ (1.07)^{10} = 1.967151 \]
4. Subtract 1 from the result:
\[ 1.967151 - 1 = 0.967151 \]
5. Divide by the interest rate \(r\):
\[ \frac{0.967151}{0.07} = 13.816442857 \]
6. Multiply by the payment amount \(PMT\):
\[ 2000 \times 13.816442857 = 27632.885714 \]
7. Round to the nearest cent:
\[ 27632.885714 \approx 27632.89 \]
Thus, the future value of the annuity is $[/tex]27,632.89.
