Sam’s grandparents deposit $250 into Sam’s savings account every month, starting from the first month after his 1st birthday. The account earns an average percentage rate of 4.8%, compounded monthly. Immediately after the last monthly deposit on Sam’s 18th birthday, the balance will be closest to[Fill in the blank]单项选择题
A
a. $78,609.88
B
b. $85,534.17
C
c. $78,924.32
D
d. $72,009.47

Question

Sam’s grandparents deposit $250 into Sam’s savings account every month, starting from the first month after his 1st birthday. The account earns an average percentage rate of 4.8%, compounded monthly. Immediately after the last monthly deposit on Sam’s 18th birthday, the balance will be closest to[Fill in the blank]单项选择题

A

a. $78,609.88

B

b. $85,534.17

C

c. $78,924.32

D

d. $72,009.47

BlackTom AI · Accepted Answer

Future value of an ordinary annuity uses FV = PMT * [((1 + i)^n − 1) / i].

1) Relevant concepts:
- Regular monthly deposits form an ordinary annuity (payments at the end of each month).
- Monthly interest rate i = APR/12.
- Number of deposits n = number of months from the first deposit (one month after 1st birthday) through the deposit on the 18th birthday.

2) Step-by-step reasoning:
- PMT = $250 per month.
- APR = 4.8% → monthly rate i = 0.048/12 = 0.004.
- First deposit at 13 months, last at 216 months (18 years = 216 months), so n = 216 − 13 + 1 = 204 deposits (equivalently 17 years × 12 = 204).

3) Calculations (all intermediate steps):
- Compute growth factor (1 + i)^n = (1.004)^204. Using a calculator:
  ln(1.004) ≈ 0.003992016 → 204·ln(1.004) ≈ 0.814371258 → (1.004)^204 ≈ e^0.814371258 ≈ 2.25775808.
- Then ((1 + i)^n − 1)/i = (2.25775808 − 1)/0.004 = 1.25775808 / 0.004 = 314.43952.
- FV = 250 × 314.43952 = $78,609.88 (rounded to cents).

4) Why this answer is correct:
- We used the correct ordinary-annuity formula, the correct monthly rate, and the correct count of monthly deposits (204). The numeric computation of (1.004)^204 gives the shown accumulation factor, producing the stated future value immediately after the last deposit.

5) Why the other multiple-choice options are incorrect:
- b. $85,534.17 is significantly larger than the correct accumulated value (would require a higher rate or more deposits).
- c. $78,924.32 is close but slightly too large; it results from a small rounding or n/i error.
- d. $72,009.47 is too small (would correspond to a lower rate or fewer deposits).