Certainly! To find the balance of an account after a certain period with a given interest rate compounded monthly, we can use the compound interest formula. The detailed step-by-step solution is as follows:
1. Identify the components:
- Initial investment ([tex]\( P \)[/tex]) = [tex]$4000
- Annual interest rate (\( r \)) = 3% = 0.03
- Compounding frequency per year (\( n \)) = 12 (since interest is compounded monthly)
- Investment duration in years (\( t \)) = 7 years
2. Write down the compound interest formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
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 interest is compounded per year.
- \( t \) is the number of years the money is invested for.
3. Substitute the values into the formula:
\[
A = 4000 \left(1 + \frac{0.03}{12}\right)^{12 \times 7}
\]
4. Calculate the inside of the parentheses first:
\[
1 + \frac{0.03}{12} = 1 + 0.0025 = 1.0025
\]
5. Raise the result to the power of \( nt \):
\[
(1.0025)^{12 \times 7} = (1.0025)^{84}
\]
6. Finally, multiply this result by the principal amount \( P \):
\[
A = 4000 \times (1.0025)^{84}
\]
7. Compute the final balance:
\[
A \approx 4933.419202196941
\]
Thus, the balance after 7 years in an account paying 3% interest compounded monthly would be approximately $[/tex]4933.42.
