If \( \$ 400 \) are deposited into an account with a \( 7 \% \) interest rate, compounded monthly, what is the balance after 8 years? \[ F=\$[?] \] \( F=P\left(1+\frac{r}{n}\right)^{n t} \)

To find the future balance, we can plug the values into the formula \( F = P \left(1 + \frac{r}{n}\right)^{nt} \). Here, \( P = 400 \), \( r = 0.07 \), \( n = 12 \) (since it's compounded monthly), and \( t = 8 \). Calculating it gives: \[ F = 400 \left(1 + \frac{0.07}{12}\right)^{12 \times 8} \] \[ F = 400 \left(1 + 0.0058333\right)^{96} \] \[ F = 400 \left(1.0058333\right)^{96} \] \[ F \approx 400 \left(1.747422\right) \] \[ F \approx 698.97 \] So, the balance after 8 years would be approximately \( F = \$698.97 \). Did you know that the concept of compound interest dates back to ancient Mesopotamia, where interest was charged on loans in grain? By the Renaissance, the idea took off in Europe, leading to complex formulas that still apply today! It's fascinating to see how something so fundamental to finance has roots in ancient civilizations. When calculating compound interest, a common mistake is forgetting to adjust the interest rate and number of periods correctly for compounding frequency. Always make sure to divide the annual interest rate by the number of compounding periods per year and multiply the number of years by that same number. This little detail can make a big difference in the final amount!