Andrew would like to accumulate $15,000 in his savings account at the end of 4 years. If effective annual interest rate is 11%. What would be the amount of equal deposit at the end of each year for 4 years.

Question

BlackTom AI · Accepted Answer

Future value of an ordinary annuity is used: S = R * [(1 + i)^n − 1] / i, where R is the end-of-year deposit, i = 0.11, n = 4, and S = 15,000.

1) Relevant concepts:
- Ordinary annuity (payments at the end of each period).
- Future value factor for an annuity: s_n = [(1 + i)^n − 1] / i.
- Solve for R by rearranging: R = S * i / [(1 + i)^n − 1].

2) Step-by-step reasoning and calculations:
- Compute (1 + i)^n = 1.11^4.
  1.11^2 = 1.2321
  1.11^3 = 1.2321 × 1.11 = 1.367631
  1.11^4 = 1.367631 × 1.11 = 1.51807041
- Compute (1 + i)^n − 1 = 1.51807041 − 1 = 0.51807041.
- Compute numerator S * i = 15,000 × 0.11 = 1,650.
- Solve for R: R = 1,650 / 0.51807041 ≈ 3,184.900 (rounded to cents $3,184.90).

3) Intermediate summary:
- The annuity future-value factor s_4 = 0.51807041 / 0.11 = 4.709731.
- Check: R × s_4 = 3,184.90 × 4.709731 ≈ 15,000 (confirms the calculation).

4) Why this answer is correct:
- Using the correct ordinary-annuity future-value formula with precise powers and arithmetic yields R ≈ $3,184.90, which produces the desired future value of $15,000 after 4 end-of-year deposits at 11% effective annual interest.

5) Why the other choices are incorrect:
- $4,355.76 and $4,834.90 produce future values well above $15,000 (too large payments).
- $2,869.28 produces a future value below $15,000 (too small payments).
- Only $3,184.90 yields the required $15,000 when multiplied by the annuity future-value factor 4.709731.

6) Conclusion:
The correct equal deposit at the end of each year is $3,184.90, matching the provided answer.

Andrew would like to accumulate $15,000 in his savings account at the end of 4 years. If effective annual interest rate is 11%. What would be the amount of equal deposit at the end of each year for 4 years.单项选择题

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