To find the amount due after 8 years under compound interest, we can use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal form).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time in years.
Let's plug in the values for this problem:
- The principal \( P \) is $2000.
- The annual interest rate \( r \) is 15%, which as a decimal is 0.15.
- The interest is compounded monthly, so \( n \) is 12.
- The time \( t \) is 8 years.
Now apply these values to the formula:
\[ A = 2000 \left(1 + \frac{0.15}{12}\right)^{12 \times 8} \]
We can calculate the interest rate per compounding period by dividing the annual interest rate by the number of compounding periods per year:
\[ \frac{0.15}{12} = 0.0125 \]
Now calculate the exponent by multiplying the number of years by the number of compounding periods per year:
\[ 12 \times 8 = 96 \]
Substitute these numbers into the compound interest formula:
\[ A = 2000 \left(1 + 0.0125\right)^{96} \]
\[ A = 2000 \left(1.0125\right)^{96} \]
To find out the numerical value of \( A \), we need to calculate \( (1.0125)^{96} \). To do this without rounding intermediate calculations, you can use a calculator:
\[ (1.0125)^{96} \approx 3.387417943 \] (without rounding the intermediate calculation)
Now multiply this by the principal:
\[ A = 2000 \times 3.387417943 \]
\[ A \approx 6774.835886 \]
Now round off the amount to the nearest penny:
\[ A \approx \$6774.84 \]
So, after 8 years, with a 15% annual interest rate compounded monthly and no payments made, the amount due would be approximately $6774.84.
