You deposit \( \$ 300 \) each month into an account earning \( 2 \% \) interest compounded monthly. a) How much will you have in the account in 25 years? b) How much total money will you put into the account? \( \$ \) c) How much total interest will you earn? \( \$ \) ?

We are given: - A monthly deposit of \( P = \$300 \). - An annual interest rate of \( 2\% \) compounded monthly. - A total time of \( 25 \) years. **Step 1. Calculate the number of compounding periods and the monthly interest rate.** The monthly interest rate is \[ r = \frac{0.02}{12}. \] The total number of deposits (periods) is \[ n = 25 \times 12 = 300. \] **Step 2. Use the future value formula for an annuity.** The formula for the future value \( A \) of an annuity (with deposits at the end of each period) is \[ A = P \cdot \frac{(1 + r)^n - 1}{r}. \] Substitute the values: \[ A = 300 \cdot \frac{\left(1 + \frac{0.02}{12}\right)^{300} - 1}{\frac{0.02}{12}}. \] **Step 3. Compute the amount in the account.** While you could leave the answer in the above form, let’s get an approximate numerical value. First, note that \[ 1 + r = 1 + \frac{0.02}{12} \approx 1.00166667. \] Thus, \[ (1 + r)^n \approx (1.00166667)^{300}. \] Since \[ (1.00166667)^{300} \approx e^{300\ln(1.00166667)} \quad \text{and} \quad \ln(1.00166667) \approx 0.00166667, \] we have approximately \[ 300 \times 0.00166667 \approx 0.5 \quad \text{so} \quad (1.00166667)^{300} \approx e^{0.5} \approx 1.64872. \] Now compute: \[ (1.64872 - 1) \approx 0.64872. \] Thus, \[ A \approx 300 \cdot \frac{0.64872}{0.00166667}. \] Calculating the denominator: \[ \frac{0.64872}{0.00166667} \approx 389.23. \] Then, \[ A \approx 300 \times 389.23 \approx 116769. \] Therefore, the amount in the account after 25 years is approximately \[ \boxed{\$116,\!770}. \] **Step 4. Calculate the total amount deposited.** Since there are \( 300 \) deposits of \( \$300 \) each, the total money deposited is \[ 300 \times 300 = \$90,\!000. \] So, \[ \boxed{\$90,\!000}. \] **Step 5. Calculate the total interest earned.** The total interest earned is the difference between the future value of the account and the total deposits: \[ \text{Interest} = A - (\text{Total Deposits}) \approx 116770 - 90000 \approx 26770. \] Thus, the total interest earned is approximately \[ \boxed{\$26,\!770}. \] **Final Answers:** a) The amount in the account in 25 years is approximately \( \$116,\!770 \). b) The total money deposited is \( \$90,\!000 \). c) The total interest earned is approximately \( \$26,\!770 \).