Lauren plans to deposit $7,000 into a bank account at the beginning of next month and $400/month into the same account at the end of that month and at the end of each subsequent month for the next 4 yr. If her bank pays interest at the rate of 5%/year compounded monthly, how much will Lauren have in her account at the end of the 4 yr? (Assume she makes no withdrawals during the 4-yr period.) Please round the answer to the nearest cent.
A) $20,794.32
B) $29,752.22
C) $35,181.39
D) $29,078.79
E) $43,898.77

Question

Lauren plans to deposit $7,000 into a bank account at the beginning of next month and $400/month into the same account at the end of that month and at the end of each subsequent month for the next 4 yr. If her bank pays interest at the rate of 5%/year compounded monthly, how much will Lauren have in her account at the end of the 4 yr? (Assume she makes no withdrawals during the 4-yr period.) Please round the answer to the nearest cent. ​
A) $20,794.32
B) $29,752.22
C) $35,181.39
D) $29,078.79
E) $43,898.77

Quizplus · Accepted Answer

Lauren's account grows through two methods: an initial lump sum deposit and regular monthly deposits. The future value of her account can be calculated using the future value formula for a lump sum and the future value formula for an annuity.1. **Future Value of the Lump Sum (Initial $7,000 Deposit):**$$FV_{lump} = P(1 + r/n)^{nt}$$Where:- $P = \$7,000$ (initial deposit),- $r = 0.05$ (annual interest rate),- $n = 12$ (compounding periods per year),- $t = 4$ (years).$$FV_{lump} = 7000(1 + 0.05/12)^{12*4}$$$$FV_{lump} = 7000(1.0041667)^{48}$$$$FV_{lump} = 7000(1.2213869)$$$$FV_{lump} ≈ \$8,549.71$$2. **Future Value of Regular Monthly Deposits:**$$FV_{annuity} = PMT \left(\frac{(1 + r/n)^{nt} - 1}{r/n}\right)$$Where:- $PMT = \$400$ (monthly deposit),- $r = 0.05$ (annual interest rate),- $n = 12$ (compounding periods per year),- $t = 4$ (years).$$FV_{annuity} = 400 \left(\frac{(1 + 0.05/12)^{12*4} - 1}{0.05/12}\right)$$$$FV_{annuity} = 400 \left(\frac{(1.0041667)^{48} - 1}{0.0041667}\right)$$$$FV_{annuity} = 400(21.5063)$$$$FV_{annuity} ≈ \$21,202.52$$3. **Total Future Value:**$$FV_{total} = FV_{lump} + FV_{annuity}$$$$FV_{total} ≈ \$8,549.71 + \$21,202.52$$$$FV_{total} ≈ \$29,752.23$$Rounding to the nearest cent, Lauren will have approximately $29,752.22 in her account at the end of the 4 years, which matches choice B.

Lauren Plans to Deposit $7,000 into a Bank Account at the Beginning