Selected terms and a rabbit–fox chase
(a) An AP has first term and difference . Its eighth, third and second terms are the first three terms of an infinite GP.
(i) Find its common ratio. marks
(ii) Find the exact sum of its odd-numbered terms in terms of . marks
(b) A programmer codes a rabbit-fox chase along a straight path to model an actual hunt. The rabbit first hops 1.75m, then 1% less each hop. The fox first leaps 3m, then 0.02m less each leap. Initially the rabbit is 60m ahead. They start and end each hop/leap at the same time.
(i) Find the total distances after leaps/hops, and the minimum number for catch-up. marks
(ii) Find fox leaps before stopping. Hence find the minimum starting distance in metres such that the fox never catches the rabbit, to the nearest integer. marks
[2022/ASRJC/Prelim/I/9]
Solution
GP ratio
Find the GP ratio from the selected AP terms.
Hint
Subtract the first two equations and then the last two.
Answer · (a)(i)
.
Write the three terms
Match AP indices 8, 3 and 2 to GP indices 1, 2 and 3, in that order.
Subtract and eliminate
The second equation equals the nonzero AP difference , so its factors are nonzero. Divide the first equation by the second to get , eliminating and together. The cancelled case would give , which the question excludes.
Odd-indexed GP sum
Find the exact infinite sum of the odd-indexed GP terms.
Hint
Odd-indexed terms start at b; even-indexed terms start at br.
Answer · (a)(ii)
.
Recover the first GP term
From d=br(1−r)=4b/25 and br²=a+d, solve for b.
Use the squared ratio
The negative sum is consistent with the negative first GP term.
First catch-up count
With a 60 m head start, find the first hop count at which the fox catches up in the endpoint model.
Hint
The rabbit ratio is 1−0.01, not 0.01.
The fox catches up when its cumulative gain reaches 60 m.
Answer · (b)(i)
, metres; first catch-up at hop .
Model each travelled distance
The rabbit multiplier is 0.99 because each hop is 1% shorter. The source margin note printing 0.01 is an error.
Translate catch-up
Compare positions by subtracting travelled distances.
Check the first crossing
Tabulate at integer indices, using the same integer-table method as in Q15. The endpoint-gain sequence increases through this first crossing; adjacent counts confirm the result.
A safe starting gap
Find the positive leaps before the fox stops, then a safe starting gap to the nearest metre.
Hint
A decreasing leap length can make the maximum gain occur before the fox stops.
The source uses endpoint/constant-progress modelling. Within-hop motion is unspecified; a strict real bound has an infimum, not an attained minimum. The source calculator’s continuous maximum is approximately 94.827 m.
Answer · (b)(ii)
positive fox leaps; safe whole-metre gap .
Endpoint/constant-progress model; real safe gaps must exceed the maximum gain ≈94.8256 m.
Stop at the last positive leap
Leap 151 has length zero, so there are 150 positive leaps.
Find the largest gain
Evaluate G over the allowed count range, not beyond the physical stopping point.
Distinguish real and whole-metre answers
The real inequality is strict; the least safe whole-metre gap is 95 m.
Explore the idea
A chase depends on cumulative gain, not the last leap
To prevent catching at any allowed endpoint, the initial gap must exceed the largest gain, not just the final gain.