Let’s solve the problem step by step: --- ### **Given:** - Regular payment, \( PMT = \$2820 \) (every 6 months) - Interest rate per year, \( r = 6\% \) compounded semiannually - Number of years, \( t = 10 \) - Payments per year = 2 (semiannual) - Total number of payments: \( n = 2 \times 10 = 20 \) --- ### **Step 1: Convert annual rate to per period (semiannual) rate** \[ i = \frac{6\%}{2} = 3\% = 0.03 \] --- ### **Step 2: Use the Ordinary Annuity Future Value Formula** \[ FV = PMT \times \frac{(1 + i)^n - 1}{i} \] - \( PMT = 2820 \) - \( i = 0.03 \) - \( n = 20 \) --- ### **Step 3: Plug in the values** \[ FV = 2820 \times \frac{(1 + 0.03)^{20} - 1}{0.03} \] --- ### **Step 4: Calculate \((1 + 0.03)^{20}\)** \[ 1.03^{20} \] First, calculate \( \ln(1.03) \approx 0.0295588 \) Then, \[ 20 \times 0.0295588 = 0.591176 \] \[ e^{0.591176} \approx 1.805 \] Or, with a calculator: \[ 1.03^{20} \approx 1.8061 \] --- ### **Step 5: Calculate the numerator** \[ 1.8061 - 1 = 0.8061 \] --- ### **Step 6: Divide by interest rate per period** \[ \frac{0.8061}{0.03} = 26.870 \] --- ### **Step 7: Multiply by Payment** \[ FV = 2820 \times 26.870 = 75,773.4 \] --- ### **Final Step: Round to the nearest cent** \[ FV \approx \$75,773.40 \] --- ## **Final Answer** **The accumulated amount in the student's annuity after 10 years will be:** \[ \boxed{\$75,773.40} \]