You plan on depositing $10,000 into a bank account each month for 3 years (36 deposits in total) with the first deposit made immediately. If the interest rate is 12% p.a. compounded monthly, how much will you have in your account in exactly 2 years and 11 months time?

Question

BlackTom AI · Accepted Answer

We’re given a stream of monthly deposits and a monthly compound interest rate, with the first deposit made immediately.
- Monthly interest rate: i = 12%/12 = 1% = 0.01.
- Timeframe: 2 years 11 months = 35 months from the start.
- Deposits: $10,000 deposited every month for 36 months, with the first deposit at time 0.
  The deposit at time 0 earns interest for 35 months, the deposit at time 1 earns for 34 months, ..., and the deposit at time 35 earns for 0 months.
  Therefore, the ending balance at t = 35 is
  Balance = 10,000 * [(1.01^0) + (1.01^1) + (1.01^2) + ... + (1.01^35)].
  This is a finite geometric series with first term a = 1 and ratio r = 1.01, so
  Sum = (r^n − 1)/(r − 1) with n = 36 terms, i.e., Sum = (1.01^36 − 1)/0.01.
  Thus, Balance = 10,000 * (1.01^36 − 1)/0.01.
  Estimating 1.01^36 ≈ e^(36 * ln 1.01) ≈ e^(36 * 0.00995) ≈ e^(0.358) ≈ 1.43, so
  (1.01^36 − 1)/0.01 ≈ (1.43 − 1)/0.01 ≈ 0.43/0.01 = 43.0.
  Therefore, Balance ≈ 10,000 * 43.0 = 430,000, which aligns with the provided options.

Evaluating each answer choice:
- $439,904.14: This is noticeably higher than the calculated sum; it would require a larger accumulated factor, suggesting an overestimate in the geometric sum or using an incorrect number of compounding periods.
- $435,076.47: Also higher than the geometric-series result; likely results from miscounting the number of deposits or the exponent applied to each deposit.
- $420,768.78: This is below the geometric-sum result; it may stem from assuming deposits start later or applying interest too few times to each deposit.
- $430,768.78: This matches the computed order of magnitude and aligns with the geometric-series formula for 36 deposits at 1% monthly over 35 months, giving a sum near 43 times the monthly deposit.
- $416,602.76: Significantly lower, which would imply far too little accumulation—likely undercounting deposits or periods.
- $435,303.52: Slightly above the geometric-sum estimate; while close, it still exceeds the precise sum derived from (1.01^36 − 1)/0.01.

Summary: The correct approach is to sum a geometric series representing each deposit’s growth to t = 35 months with 1% monthly interest, yielding approximately 430,000. Among the options, $430,768.78 best matches that precise calculation.

You plan on depositing $10,000 into a bank account each month for 3 years (36 deposits in total) with the first deposit made immediately. If the interest rate is 12% p.a. compounded monthly, how much will you have in your account in exactly 2 years and 11 months time?单项选择题

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