An account is opened with an initial deposit of \( \$ 100 \) and earns \( 3.0 \% \) interest compounded monthly. What will the account be worth in 25 years? Round your answer to the nearest cent. Do NOT round until you have calculated the final answer.

To determine the future value of an account with an initial deposit of \$100 earning 3.0% interest compounded monthly over 25 years, we'll use the **compound interest formula**: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) = the amount of money accumulated after \( t \) years, including interest. - \( P \) = the principal amount (\$100). - \( r \) = annual interest rate (decimal) (3.0% or 0.03). - \( n \) = number of times that interest is compounded per year (12 for monthly). - \( t \) = the time the money is invested for in years (25). Let's plug in the values: \[ A = 100 \left(1 + \frac{0.03}{12}\right)^{12 \times 25} = 100 \left(1 + 0.0025\right)^{300} = 100 \times (1.0025)^{300} \] **Calculating \((1.0025)^{300}\):** To compute \((1.0025)^{300}\), we'll use the natural logarithm and exponential functions for higher precision. 1. **Calculate the natural logarithm:** \[ \ln(1.0025) \approx 0.00249688 \] 2. **Multiply by 300:** \[ 300 \times 0.00249688 = 0.749064 \] 3. **Exponentiate to find the factor:** \[ e^{0.749064} \approx 2.11502 \] **Final Calculation:** \[ A = 100 \times 2.11502 = 211.502 \] Rounding to the nearest cent: \[ A \approx \$211.50 \] **Answer:** After 25 years, the account will be worth **\$211.50**.