Questions: Kurtiss has a client who wants to invest in an account that earns 6% 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 fifth deposit is a.) 20,288 b.) 21,097 c.) 22,548 d.) 23,075

Kurtiss has a client who wants to invest in an account that earns 6% 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 fifth deposit is
a.) 20,288
b.) 21,097
c.) 22,548
d.) 23,075

Transcript text: Kurtiss has a client who wants to invest in an account that earns 6% 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 fifth deposit is a.) $20,288 b.) $21,097 c.) $22,548 d.) $23,075

Solution

Solution Steps

Step 1: Calculate Future Value of Initial Deposit

The future value $ FV_P $ of the initial deposit after 5 years can be calculated using the formula: \[ FV_P = P \times (1 + r)^n \] Substituting the values: \[ FV_P = 4000 \times (1 + 0.06)^5 = 4000 \times (1.338225) \approx 5352.90 \]

Step 2: Calculate Future Value of Annual Deposits

The future value $ FV_D $ of the annual deposits can be calculated using the formula: \[ FV_D = D \times \left(\frac{(1 + r)^n - 1}{r}\right) \] Substituting the values: \[ FV_D = 4000 \times \left(\frac{(1 + 0.06)^5 - 1}{0.06}\right) = 4000 \times \left(\frac{1.338225 - 1}{0.06}\right) \approx 4000 \times 5.637083 \approx 22548.33 \]

Step 3: Calculate Total Future Value

The total future value $ FV $ of the account after 5 years is the sum of the future values of the initial deposit and the annual deposits: \[ FV = FV_P + FV_D \approx 5352.90 + 22548.33 \approx 27901.27 \]