To find the balance of an account after a certain period given a principal amount, an interest rate, and compound frequency, you can use the compound interest formula:
[tex]\[
F = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
where:
- [tex]\( F \)[/tex] is the future value of the investment/loan, including interest
- [tex]\( P \)[/tex] is the principal investment amount (initial deposit)
- [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 time the money is invested or borrowed for, in years
Given the parameters:
- Principal [tex]\( P = \$ 850 \)[/tex]
- Annual interest rate [tex]\( r = 0.06 \)[/tex] (6\%)
- Number of times compounded per year [tex]\( n = 12 \)[/tex] (monthly)
- Time [tex]\( t = 7 \)[/tex] years
Substitute these values into the compound interest formula:
[tex]\[
F = 850 \left(1 + \frac{0.06}{12}\right)^{12 \times 7}
\][/tex]
First, calculate the monthly interest rate:
[tex]\[
\frac{0.06}{12} = 0.005
\][/tex]
Next, determine the total number of compounding periods over 7 years:
[tex]\[
12 \times 7 = 84
\][/tex]
Now substitute these values into the expression:
[tex]\[
F = 850 \left(1 + 0.005\right)^{84}
\][/tex]
Evaluate the term inside the parentheses:
[tex]\[
1 + 0.005 = 1.005
\][/tex]
Raise [tex]\( 1.005 \)[/tex] to the power of 84:
[tex]\[
(1.005)^{84}
\][/tex]
Now multiply this result by the principal amount [tex]\( 850 \)[/tex]:
[tex]\[
F = 850 \times (1.005)^{84} = 850 \times 1.52027559 \approx 1292.31
\][/tex]
Therefore, the balance after 7 years, rounded to the nearest cent, is:
[tex]\[
F = \$ 1292.31
\][/tex]
