Let-the-first-term-and-the-common-ratio-of-a-geometric-sequence-a-n-be-1-and-r-If-a-n-satisfy-a-n-1-a-1-a-n-a-1-for-all-n-2-find-the-range-of-values-of-r-

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Question Number 108805 by ZiYangLee last updated on 19/Aug/20

Let the first term and the common ratio of a geometric sequence {a_n } be 1 and r. If {a_n } satisfy ∣a_(n−1) −a_1 ∣≤∣a_n −a_1 ∣ for all n≥2, find the range of values of r.

$$\mathrm{Let}\:\mathrm{the}\:\mathrm{first}\:\mathrm{term}\:\mathrm{and}\:\mathrm{the}\:\mathrm{common} \\ $$$$\mathrm{ratio}\:\mathrm{of}\:\mathrm{a}\:\mathrm{geometric}\:\mathrm{sequence}\:\left\{{a}_{\mathrm{n}} \right\}\:\mathrm{be}\:\mathrm{1} \\ $$$$\mathrm{and}\:{r}.\: \\ $$$$\mathrm{If}\:\left\{{a}_{\mathrm{n}} \right\}\:\mathrm{satisfy}\:\mid{a}_{\mathrm{n}−\mathrm{1}} −{a}_{\mathrm{1}} \mid\leqslant\mid{a}_{\mathrm{n}} −{a}_{\mathrm{1}} \mid\:\mathrm{for} \\ $$$$\mathrm{all}\:\mathrm{n}\geqslant\mathrm{2},\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{values}\:\mathrm{of}\:{r}. \\ $$

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Let-the-first-term-and-the-common-ratio-of-a-geometric-sequence-a-n-be-1-and-r-If-a-n-satisfy-a-n-1-a-1-a-n-a-1-for-all-n-2-find-the-range-of-values-of-r-

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