Question 0414

Arithmetic and Geometric Progressions (APs, GPs)

2014 Paper 2 Question 3 Variant

Question

In a training exercise, athletes run from a starting point

{O}

to and from a series of points,

{A_1,}

{A_2,}

{A_3, \ldots,}

increasingly far away in a straight line. In the exercise, athletes start at

{O}

and run stage

{1}

from

{O}

{A_1}

and back to

{O,}

then stage

{2}

from

{O}

{A_2}

and back to

{O,}

and so on.

(i)

In Version

{1}

of the exercise, the distance between adjacent points are all

{3}

(ia)

Find the distance run by an athlete who completes the first

{30}

stages of Version

{1}

of the exercise.

[2]

(ib)

Write down an expression for the distance run by an athlete who completes

{n}

stages of Version

{1.}

Hence find the least number of stages that the athlete needs to complete to run at least

{7}

km.

[4]

(ii)

In Version

{2}

of the exercise, the distance between the points are such that

{OA_1 = 3}

{A_1A_2 = 3}

{A_2A_3 = 6}

m and

{A_{n}A_{n+1}=2A_{n-1}A_n.}

Write down an expression for the distance run by an athlete who completes

{n}

stages of Version

{2.}

Hence find the distance from

{O,}

and the direction of travel, of the athlete after he has run exactly

{14}

km, using Version

{2.}

[5]