To determine the future value of an investment under compound interest, we use the compound interest formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( n \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [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 number of years the money is invested for.
Given values:
- [tex]\( P = 300 \)[/tex] (the principal investment amount)
- [tex]\( r = 0.05 \)[/tex] (5% annual interest rate)
- [tex]\( n = 4 \)[/tex] (interest is compounded quarterly, so 4 times per year)
- [tex]\( t = 20 \)[/tex] years
Now, let's use the given values in the formula:
[tex]\[ A = 300 \left(1 + \frac{0.05}{4}\right)^{4 \times 20} \][/tex]
First, calculate the term inside the parentheses:
[tex]\[ \frac{0.05}{4} = 0.0125 \][/tex]
So, the expression becomes:
[tex]\[ A = 300 \left(1 + 0.0125\right)^{4 \times 20} \][/tex]
Simplify the exponent:
[tex]\[ 4 \times 20 = 80 \][/tex]
Now the formula looks like this:
[tex]\[ A = 300 \left(1 + 0.0125\right)^{80} \][/tex]
Calculate the inside of the parentheses:
[tex]\[ 1 + 0.0125 = 1.0125 \][/tex]
Thus, the expression becomes:
[tex]\[ A = 300 \left(1.0125\right)^{80} \][/tex]
Now, raise 1.0125 to the power of 80:
[tex]\[ 1.0125^{80} \approx 2.70148494075 \][/tex]
Finally, multiply by the principal amount:
[tex]\[ A = 300 \times 2.70148494075 \approx 810.4454822259983 \][/tex]
Thus, the investment will be worth approximately:
[tex]\[ \$810.45 \][/tex]
Of the given options, the correct answer is:
\$810.45
