To determine the future value of an investment with compounded interest, we utilize the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the future value of the investment/loan, including interest.
- [tex]\( P \)[/tex] is the principal investment amount (the initial deposit or loan amount).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested or borrowed for, in years.
Given:
- The principal amount, [tex]\( P = 26000 \)[/tex] dollars.
- The annual interest rate, [tex]\( r = 0.08 \)[/tex] (which is 8%).
- The investment period, [tex]\( t = 8 \)[/tex] years.
- The interest is compounded monthly, hence [tex]\( n = 12 \)[/tex].
Step-by-step calculation:
1. Convert the annual interest rate to a decimal: [tex]\( r = 8\% = 0.08 \)[/tex].
2. Calculate the monthly interest rate:
[tex]\[
\frac{r}{n} = \frac{0.08}{12} \approx 0.0066667
\][/tex]
3. Determine the total number of compounding periods:
[tex]\[
nt = 12 \times 8 = 96
\][/tex]
4. Substitute the values into the compound interest formula:
[tex]\[
A = 26000 \left(1 + \frac{0.08}{12}\right)^{12 \times 8}
\][/tex]
[tex]\[
A = 26000 \left(1 + 0.0066667\right)^{96}
\][/tex]
[tex]\[
A = 26000 \left(1.0066667\right)^{96}
\][/tex]
5. Compute the expression:
[tex]\[
A \approx 26000 \times 1.892458
\][/tex]
[tex]\[
A \approx 49203.89
\][/tex]
So, the future value of the investment rounded to two decimal places is [tex]\(\$49,203.89\)[/tex].
Thus, after 8 years, the investment will grow to approximately [tex]\(\$49,203.89\)[/tex].
