Paula has a client who wants to invest into an account that earns \( 4 \% \) interest, compounded annually. The client opens the account with an initial deposit of \( \$ 4,000 \), and deposits an additional \( \$ 4,000 \) into the account each year thereafter. Assuming no withdrawals or other deposits are made and that the interest rate is fixed, the balance of the account (rounded to the nearest dollar) after the eighth deposit is a.) \( \$ 188,247 \) b.) \( \$ 40,969 \) c.) \( \$ 102,713 \) d.) \( \$ 36,857 \) (

To find the balance in the account after the eighth deposit, we can use the formula for the future value of a series of cash flows compounded annually. The formula for the future value of an annuity compounded annually is given by: \[ FV = P \times \frac{(1 + r)^n - 1}{r} + P_0 \times (1 + r)^n \] Where: - \( P \) is the annual deposit (\$4,000) - \( r \) is the annual interest rate (0.04) - \( n \) is the number of deposits (8) - \( P_0 \) is the initial deposit (\$4,000) Substituting in the values, we have: 1. **Future Value of Annual Deposits:** \[ FV_{\text{deposits}} = 4000 \times \frac{(1 + 0.04)^8 - 1}{0.04} \] \[ = 4000 \times \frac{(1.36856962 - 1)}{0.04} \approx 4000 \times 9.21424 \approx 36,857 \] 2. **Future Value of the Initial Deposit:** \[ FV_{\text{initial}} = 4000 \times (1 + 0.04)^8 \] \[ = 4000 \times 1.36856962 \approx 5,474 \] Adding these amounts together gives us the total future value: \[ Total FV = 36,857 + 5,474 \approx 42,331 \] Now, since we are looking for the balance after the eighth deposit, we compute the total balance as: 1. **Final Calculation**: Total balance after eight deposits, using the previously found values equals approximately \$42,331. Based on your options: None of the options given were close to \$42,331. However, please ensure the calculations are appropriately done or based on an understanding prior to it presented here. To cycle back to the given options, consistency in categorization of financial growth is key to reaching the correct value based on initial conditions before the eighth contribution begins compounding. Thus revisions in foundational aspects of financial structuring could yield refined estimations.