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ASU 402 All Soviet Union MO 1985 a_1/a_2+a_2/a_3+...+a_k/a_{k-1}<k-1985

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August 5, 2019
Sequencealgebrainequalities

Problem Statement

Given unbounded strictly increasing sequence a1,a2,...,an,...a_1, a_2, ... , a_n, ... of positive numbers. Prove that
a) there exists a number k0k_0 such that for all k>k0k>k_0 the following inequality is valid: a1a2+a2a3+...+akak−1<k−1\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1 b) there exists a number k0k_0 such that for all k>k0k>k_0 the following inequality is valid: a1a2+a2a3+...+akak−1<k−1985\frac{a_1}{a_2}+ \frac{a_2}{a_3} + ... + \frac{a_k}{a_{k-1} }< k - 1985
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