To determine the future value of an investment with compound interest, we will use the compound interest formula, which is:
[tex]\[ A = P(1 + r)^n \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (in decimal form).
- [tex]\( n \)[/tex] is the number of years the money is invested or borrowed for.
Given the values:
- Principal amount ([tex]\( P \)[/tex]) = \[tex]$800
- Annual interest rate (\( r \)) = 9.5% or 0.095 in decimal form
- Number of years (\( n \)) = 3
Let's substitute the given values into the formula:
\[ A = 800 \left(1 + 0.095\right)^3 \]
Now, calculate the expression inside the parentheses first:
\[ 1 + 0.095 = 1.095 \]
Next, raise this result to the power of 3:
\[ \left(1.095\right)^3 \approx 1.311299 \]
Finally, multiply this value by the principal amount:
\[ A = 800 \times 1.311299 \approx 1049.0392 \]
Thus, the final value of the investment after 3 years, compounded annually at an interest rate of 9.5%, is:
\[ A \approx 1050.3459 \]
So, the value of the investment after 3 years would be approximately \$[/tex]1050.35.
