Yuxuan puts [math: $32000] $32\hspace{1mm} 000 in an account earning [math: 6.1%] 6.1\,\% interest per annum, compounding fortnightly. The interest is charged at the end of every fortnight and right after that he regularly adds [math: $140] to his investment. Which of the following is closest to the value of Yuxuan’s investment at the end of the fourteenth fortnight before he adds the [math: $140] ?[Fill in the blank]

Question

BlackTom AI · Accepted Answer

We start by identifying the renewal process each fortnight. The account balance earns interest for the fortnight at a rate i = 6.1% per year compounded fortnightly, so the per-fortnight rate is i = 0.061 / 26 ≈ 0.002346154. After earning interest for the fortnight, $140 is added. Therefore, if B_k is the balance after the k-th fortnight (including the $140 addition), the recurrence is B_k = B_{k-1}*(1 + i) + 140, with B_0 = 32000.

Option A. $35,057
To assess this, one would need to compute B_14 before the 14th addition, i.e., the balance just after interest is applied in fortnight 14 but before adding the $140. This value is B_13*(1+i). Quick estimation shows this value is unlikely to be as high as 35,057 given the early base of 32,000 and only modest fortnightly growth; however, precise calculation is required to confirm. Without performing the full calculation, it’s not obviously correct or incorrect.

Option B. $34,917
This value is very close to a careful calculation. Using the closed-form solution for the recurrence, B_k = B_0*(1+i)^k + 140*[( (1+i)^k - 1 ) / i]. The balance just before the 14th addition is B_13*(1+i). Plugging B_0 = 32000, i ≈ 0.002346154, and k = 13, and then multiplying by (1+i), yields a result near 34,917. The arithmetic aligns with the expected scale of growth (a bit over 34k). This option is plausible.

Option C. $36,814
This value is notably higher than what the investment would reach after 13 fortnights with such a small fortnightly rate, even before adding the 14th amount. It would correspond to either a much larger effective growth (higher i) or many more fortnights, neither of which applies here. Therefore this is unlikely.

Option D. $37,120
Similar to option C, this is also well above the calculated range for 14 fortnights with a 0.61% annual rate compounded fortnightly. The sequence we’re following grows slowly each fortnight, and reaching over 37k before the 14th addition would require substantially more compounding or a larger contribution, which is not present here. This is unlikely.

Summary of reasoning: The correct approach is to use the closed-form for the recurrence B_k = B_0*(1+i)^k + 140*((1+i)^k - 1)/i to get B_13, then multiply by (1+i) to obtain the value just before the 14th addition. Doing the calculation yields a result around 34,917, which makes option B the closest estimate. The other options either overestimate (C and D) or require more precise computation to verify but are not as close to the computed value as B.

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