Theory question 109
Sequences and Series(a) A jogger is training for a 15 km Charity race. He starts with a run of 500 metres, then he increases the distance he runs daily by 250 metres. (i) How many days will it take the jogger to reach a distance of 15 km in training? (ii) Calculate the total distance he would have run in the training. (b) The second term of a Geometric Progrssion (G.P.) is −3. If its sum to infinity is 25/2, find its common ratios.
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(a)(i) 500 + 250(n − 1) = 15 000 gives n − 1 = 58, n = 59 days.
(ii) S = 59/2 × (500 + 15 000) = 457 250 m = 457.25 km.
(b) ar = −3 and a/(1 − r) = 25/2. With a = −3/r: −6 = 25r(1 − r), so 25r² − 25r − 6 = 0, (5r + 1)(5r − 6) = 0.
r = −1/5 or 6/5; a sum to infinity needs |r| < 1, so r = −1/5 (a = 15).
A sum to infinity exists only when −1 < r < 1.