To determine how much you would need to deposit into an account with a 6% annual interest rate, compounded quarterly, to have [tex]$2250 in your account 17 years later, we can use the formula for compound interest, which is:
\[
F = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( F \) is the future value ($[/tex]2250 in this case)
- [tex]\( P \)[/tex] is the principal amount (the amount to be determined)
- [tex]\( r \)[/tex] is the annual interest rate (6% or 0.06 in decimal form)
- [tex]\( n \)[/tex] is the compounding frequency (quarterly compounding means [tex]\( n = 4 \)[/tex])
- [tex]\( t \)[/tex] is the number of years the money is invested (17 years)
First, we need to rearrange the formula to solve for [tex]\( P \)[/tex]:
[tex]\[
P = \frac{F}{\left(1 + \frac{r}{n}\right)^{nt}}
\][/tex]
Substitute the given values into the formula:
[tex]\[
P = \frac{2250}{\left(1 + \frac{0.06}{4}\right)^{4 \cdot 17}}
\][/tex]
Let's break this down into smaller steps:
1. Calculate the quarterly interest rate:
[tex]\[
\frac{0.06}{4} = 0.015
\][/tex]
2. Add 1 to the quarterly interest rate:
[tex]\[
1 + 0.015 = 1.015
\][/tex]
3. Raise this amount to the power of the total number of compounding periods (which is [tex]\( 4 \times 17 \)[/tex]):
[tex]\[
1.015^{68}
\][/tex]
4. Divide the future value by this amount to find the present value [tex]\( P \)[/tex]:
[tex]\[
P = \frac{2250}{1.015^{68}}
\][/tex]
After performing the calculations, we find:
[tex]\[
P \approx 817.5073108142155
\][/tex]
Rounding to the nearest cent, we get:
[tex]\[
P \approx 817.51
\][/tex]
So, you would have to deposit approximately [tex]$817.51 into the account to have $[/tex]2250 in your account 17 years later with a 6% interest rate compounded quarterly.
