To find out how much you will have in the account after 15 years with an initial deposit of [tex]$4500, an annual interest rate of 5.2%, and monthly compounding, we can follow the steps of the compound interest formula. The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested for.
Let's break down the steps:
1. Principal (P): The initial deposit amount, which is $[/tex]4500.
2. Annual interest rate (r): The annual interest rate is 5.2%. To express it as a decimal, we divide by 100:
[tex]\[ r = \frac{5.2}{100} = 0.052 \][/tex]
3. Number of times interest is compounded per year (n): Since the interest is compounded monthly, [tex]\( n = 12 \)[/tex].
4. Number of years (t): The time period the money is invested for, which is 15 years.
5. Plugging these values into the formula:
[tex]\[
A = 4500 \left(1 + \frac{0.052}{12}\right)^{12 \times 15}
\][/tex]
6. Calculate the monthly interest rate:
[tex]\[
\frac{0.052}{12} = 0.004333
\][/tex]
7. Calculate the exponent [tex]\( 12 \times 15 = 180 \)[/tex].
8. Add 1 to the monthly interest rate:
[tex]\[
1 + 0.004333 = 1.004333
\][/tex]
9. Raise the result to the power of 180:
[tex]\[
1.004333^{180}
\][/tex]
10. Multiply by the principal amount ([tex]$4500):
\[
4500 \times 1.004333^{180} \approx 9800.0968
\]
After following these steps, the amount in the account after 15 years will be approximately $[/tex]9800.10.
