Consider the geometric progression whose first term is 7 and whose common ratio is

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Consider the geometric progression whose first term is 7 and whose common ratio is −12-\dfrac12. For positive integers nn, let SnS_n be the sum of the first nn terms. (i) Obtain a simplified expression for SnS_n in terms of nn. (ii) Show that S2n=143 (1−(−12)2n)S_{2n}=\dfrac{14}{3}\,\bigl(1-\bigl(-\dfrac12\bigr)^{2n}\bigr). (iii) Using your result from part (i), find the least integer nn such that ∣S∞−Sn∣<0.0001\lvert S_\infty-S_n\rvert<0.0001, where S∞S_\infty denotes the sum to infinity of the progression. Present a clear justification for your answer.

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