To find the future value of an investment compounded quarterly, 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]\( t \)[/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 interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested or borrowed for, in years.
Given:
- Principal ([tex]\( P \)[/tex]) = \[tex]$360
- Annual interest rate (\( r \)) = 4% or 0.04 (as a decimal)
- Number of times interest is compounded per year (\( n \)) = 4 (quarterly)
- Number of years (\( t \)) = 18
Let's substitute these values into the formula:
\[ A = 360 \left(1 + \frac{0.04}{4}\right)^{4 \times 18} \]
First, compute the term inside the parentheses:
\[ \frac{0.04}{4} = 0.01 \]
Add 1 to this value:
\[ 1 + 0.01 = 1.01 \]
Raise this value to the power of \( 4 \times 18 \):
\[ (1.01)^{72} \]
Now, multiply this result by the principal amount:
\[ A = 360 \times (1.01)^{72} \]
Given the final computation:
\[ A \approx 736.96 \]
Thus, the investment will be worth approximately \$[/tex]736.96 after 18 years when compounded quarterly at an annual interest rate of 4%.
The correct answer is:
[tex]\[\$ 736.96\][/tex]
