To find out how much Beth will have at the end of 9 years by depositing $2000 at the end of each six-month period in an account that pays 8% interest, compounded semiannually, we can use the future value of an ordinary annuity formula:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) = future value of the annuity
- \( P \) = payment amount per period
- \( r \) = interest rate per period
- \( n \) = total number of payments
Given:
- \( P = 2000 \)
- Annual interest rate = 8%, so the semiannual interest rate \( r = \frac{0.08}{2} = 0.04 \)
- Total time = 9 years, with 2 periods per year, gives us \( n = 9 \times 2 = 18 \)
Now, plug the values into the formula:
\[ FV = 2000 \times \frac{(1 + 0.04)^{18} - 1}{0.04} \]
Calculating \( (1 + 0.04)^{18} \):
\[ (1 + 0.04)^{18} = (1.04)^{18} \approx 1.868246 \]
Now substitute back:
\[ FV = 2000 \times \frac{1.868246 - 1}{0.04} \] \[ FV = 2000 \times \frac{0.868246}{0.04} \] \[ FV = 2000 \times 21.70615 \approx 43412.30 \]
Thus, the total amount that Beth will have at the end of 9 years is approximately $43,412.30.