N2014 P2 Q3

In a training exercise, athletes run from a starting point \(O\) to and from a series of points, \({{A}_{1}}\), \({{A}_{2}}\), \({{A}_{3}}\),….., increasingly far away in a straight line. In the exercise, atheletes start at \(O\) and run stage 1 from \(O\) to \({{A}_{1}}\) and back to \(O\), then stage 2 from \(O\) to \({{A}_{2}}\) and back to \(O\), and so on.

1. In Version 1 of the exercise, the distances between adjacent points are all \(4\)m (see Fig. 1).
   
   1. Find the distance run by an athlete who completes the first \(10\) stages of Version 1 of the exercise.[2]
   2. Write down an expression for the distance run by an athlete who completes \(n\) stages of Version I. Hence find the least number of stages that the athlete needs to complete to run at least \(5\)km.[4]
2. In Version 2 of the exercise, the distances between the points are such that \(O{{A}_{1}}=4\)m, \({{A}_{1}}{{A}_{2}}=4\)m, \({{A}_{2}}{{A}_{3}}=8\)m, \({{A}_{n}}{{A}_{n+1}}=2{{A}_{n-1}}{{A}_{n}}\) (see Fig. 2). 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 \(10\text{ km}\) using Version 2.[5]