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Worked solution

Add a late fee after interest and count the charging cycles

A man enters a coma after an accident on 10 January 2013 and misses payment of his credit-card balance M dollars due on 27 January. On each monthly 27th, unpaid balances first receive 2%2\% interest and then an additional late fee L dollars.

(a) Express his balance on 1 February 2013 in terms of M and L. mark

(b) Show that exactly n months after the accident, on the same day of the month, the accumulated balance is (1.02)nM+50L[(1.02)n−1](1.02)^nM+50L[(1.02)^n-1]. marks

(c) For M=1000M=1000, L=55L=55, find the least n for which the balance first exceeds $2010. marks

[2013/AJC/Promo/15]

CJC · Review problems · Question 22

Answer

(a) 1.02M+L1.02M+L; (b) (1.02)nM+50L[(1.02)n−1](1.02)^nM+50L[(1.02)^n-1]; (c) least n=13n=13.

(a) Apply one interest-and-fee cycle

The first charge occurs on 27 January. By 1 February the balance is 1.02M+L1.02M+L dollars. The late fee is added after interest, so it is not multiplied by 1.021.02 within that same cycle.

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(b) Retain the cycle table and the geometric fee contributions

For integer n, the observation on the 10th n months later includes n completed 27th-of-month charges. Let B0=MB_0=M, Bn=1.02Bn−1+LB_n=1.02B_{n-1}+L.

CyclesBalance after the charge
11.02M+L1.02M+L
2(1.02)2M+L(1.02+1)(1.02)^2M+L(1.02+1)
3(1.02)3M+L[(1.02)2+1.02+1](1.02)^3M+L[(1.02)^2+1.02+1]
4(1.02)4M+L[(1.02)3+(1.02)2+1.02+1](1.02)^4M+L[(1.02)^3+(1.02)^2+1.02+1]
n(1.02)nM+L∑j=0n−1(1.02)j(1.02)^nM+L\sum_{j=0}^{n-1}(1.02)^j

Summing the fee GP gives Bn=(1.02)nM+L[(1.02)n−1]/0.02=(1.02)nM+50L[(1.02)n−1]B_n=(1.02)^nM+L[(1.02)^n-1]/0.02=(1.02)^nM+50L[(1.02)^n-1].

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(c) Solve the strict inequality and check the neighbouring integers

For the stated values, Bn=3750(1.02)n−2750B_n=3750(1.02)^n-2750. Thus Bn>2010B_n>2010 exactly when (1.02)n>476/375(1.02)^n>476/375, or n>ln⁡(476/375)/ln⁡1.02≈12.04n>\ln(476/375)/\ln1.02\approx12.04. The least integer is 1313. Indeed B12≈2005.91B_{12}\approx2005.91 while B13≈2101.02B_{13}\approx2101.02, confirming the threshold. These are integer monthly observations; the recurrence is not a continuous daily-interest model.

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Common mistakes

Adding the fee before multiplying by 1.02 would charge interest on that new fee immediately.

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