Porisma

Worked solution

Count elapsed updates separately from the week label

An insect colony initially has 5050 insects. Each week its population doubles, then 3030 die through competition and limited resources. Let unu_n be the population at the beginning of week nn, n=1,2,…n=1,2,\ldots, with all remaining insects alive.

(i) Form a recurrence relating week n and week n−1. mark

(ii) Find the minimum number of weeks needed for the population to exceed 300000300000. marks

[2006/MJC/Promo/2(b)]

CJC · Review problems · Question 14

Answer

un=2un−1−30u_n=2u_{n-1}-30, u1=50u_1=50 for beginning-of-week labels; threshold after 1414 completed weeks, at the beginning of week 1515.

(i) State the recurrence with an explicit initial index

Each update is un=2un−1−30u_n=2u_{n-1}-30. With the prompt’s beginning-of-week labels, u1=50u_1=50 and the recurrence applies for n≥2n\ge2. The source instead uses u0=50u_0=50, treating the subscript as the number of elapsed weekly updates; both conventions describe the same physical process after shifting the index.

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(ii) Use the source elapsed-update convention

Let vmv_m be the population after mm completed weeks, so v0=50v_0=50 and vm=2vm−1−30v_m=2v_{m-1}-30. Then vm−30=2(vm−1−30)v_m-30=2(v_{m-1}-30), hence vm=30+20⋅2mv_m=30+20\cdot2^m.

Completed weeks mPopulation v_m
13163870
14327710
15655390

The population first exceeds 300000300000 after 14 completed weeks, as reported in the source. Under the prompt’s week-one initial label, this is the beginning of week 1515.

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Common mistakes

The source initial index is zero, while the prompt calls the initial population week one.

Read the hint