An arithmetic progression has first term $a$ and common difference $d$, where $a0$ and $d
e 0$. Th

Question

An arithmetic progression has first term $a$ and common difference $d$, where $a>0$ and $d
e 0$. The eighth, third and second term of the progression are the first three terms of an infinite geometric progression. Find the common ratio of the geometric progression. [3] Find the exact sum of the odd-numbered terms of the geometric progression in terms of $a$. [3] A programmer coded a program involving a rabbit-fox chase along a straight path to model the actual hunt for a rabbit by a fox. The rabbit first hop is $1.75$m. In each subsequent hop, the distance covered is $1\%$ less than its previous hop. The fox first leaps $3$ m. In each subsequent leap, the distance covered is $0.02$m less than its previous leap. Initially the rabbit is $60$ m ahead of the fox and assume that the rabbit and the fox start and end each hop and leap at the same time. By finding the total distance travelled by the fox and the rabbit after n leaps and hops respectively, find the minimum number of hops and leaps for the fox to catch up with the rabbit. [4] Find the number of leaps the fox takes before it comes to a stop. Hence, find the minimum starting distance, in metre, between the fox and the rabbit such that the fox will never catch up with the rabbit. Leave your answer to the nearest integer. [2]

Accepted Answer

(a)(i) $r=\frac15$ (a)(ii) $-\frac{625}{72}a$ (b)(i) $54$ (b)(ii) $150$ leaps, minimum starting distance $=95\text{ m}$

2022 ASRJC P1 Q9

2022 ASRJC P1 Q9

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