How long will it take for weekly deposits of $100 to amount to $10 768 if the interest rate is 3.5% per annum compounded weekly?

Question

BlackTom AI · Accepted Answer

Future value of a stream of weekly deposits uses the ordinary annuity formula: FV = PMT * [((1 + i)^n - 1) / i], where PMT = 100, i = periodic (weekly) interest rate, and n = number of weekly deposits.

1) Relevant concepts
- Convert annual nominal rate to the weekly periodic rate: i = 0.035 / 52.
- Use the future value formula for regular deposits (ordinary annuity).
- Solve for n (number of weeks) by isolating (1 + i)^n and taking logarithms, then convert weeks to years (divide by 52).

2) Step-by-step reasoning and calculations
- Compute weekly rate: i = 0.035 / 52 = 0.0006730769231.
- Set up equation with target FV = 10,768:
  10,768 = 100 * [((1 + i)^n - 1) / i].
- Rearrange:
  (1 + i)^n - 1 = (10,768 * i) / 100.
- Compute the right-hand side:
  10,768 * i = 10,768 * (0.035 / 52) = (10,768 * 0.035) / 52 = 376.88 / 52 = 7.2484615385.
  Divide by 100: (10,768 * i) / 100 = 7.2484615385 / 100 = 0.072484615385.
- So (1 + i)^n = 1 + 0.072484615385 = 1.072484615385.
- Take natural logs to solve for n:
  n = ln(1.072484615385) / ln(1 + i).
  Compute numerator: ln(1.072484615385) ≈ 0.0698556.
  Compute denominator: ln(1 + i) ≈ ln(1.0006730769231) ≈ 0.00067285.
  Thus n ≈ 0.0698556 / 0.00067285 ≈ 103.83 weeks.
- Convert weeks to years: 103.83 / 52 ≈ 1.9975 years ≈ 2 years.

3) Why the correct answer is correct
The calculation yields approximately 103.8 weekly deposits, which is essentially 104 weeks or 2 years. That exactly matches option A.

4) Why other options are incorrect
- B (1 year): 52 weekly deposits (without interest $5,200; with small weekly interest only slightly higher) is far below $10,768.
- C (1 1/2 years = 78 weeks): 78*100 = $7,800 (plus interest still well below $10,768).
- D (3 years = 156 weeks): 156*100 = $15,600 (well above $10,768).

Conclusion: The correct choice is A. 2 years, which agrees with the provided answer.

How long will it take for weekly deposits of $100 to amount to $10 768 if the interest rate is 3.5% per annum compounded weekly?单项选择题

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