Certainly! To find out how much an investment of [tex]$3000 at an annual interest rate of 7.75%, compounded annually, will be worth after 14 years, we use the compound interest formula. Here is the step-by-step solution:
The compound interest formula is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for in years.
In this problem:
- \( P = 3000 \) dollars
- \( r = 7.75\% = 0.0775 \) (convert percentage to decimal)
- Since the interest is compounded annually, \( n = 1 \)
- \( t = 14 \) years
Given that the interest is compounded annually, the formula simplifies to:
\[ A = P \left(1 + r\right)^t \]
Now we substitute the given values into the formula:
\[ A = 3000 \left(1 + 0.0775\right)^{14} \]
Calculate the term inside the parentheses first:
\[ 1 + 0.0775 = 1.0775 \]
Raise this value to the power of 14:
\[ 1.0775^{14} \approx 2.9527 \]
Multiply by the principal amount:
\[ A = 3000 \times 2.9527 \approx 8858.1 \]
Finally, round the result to the nearest dollar:
\[ A \approx 8858 \]
So, the investment will be worth approximately $[/tex]8,858 after 14 years.
