In a training exercise, athletes run from a starting point $O$ to and from a series of points, ${{A}

Question

In a training exercise, athletes run from a starting point $O$ to and from a series of points, ${{A}_{1}}$, ${{A}_{2}}$, ${{A}_{3}}$,….., 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. In Version 1 of the exercise, the distances between adjacent points are all $4$m (see Fig. 1). Find the distance run by an athlete who completes the first $10$ stages of Version 1 of the exercise.[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] 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.

Accepted Answer

(i)(a) $440\text{ m}$; (i)(b) 35 stages; (ii) $S_n=8(2^n-1)$, after the 10th stage the athlete is 1816 m from $A_{11}$

N2014 P2 Q3

N2014 P2 Q3

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