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recurrence inequality, sequence is not all-positive

Source: Bulgaria 1986 P6

June 15, 2021
inequalitiesrecurrence relationSequencesalgebra

Problem Statement

Let 0<k<10<k<1 be a given real number and let (an)n≥1(a_n)_{n\ge1} be an infinite sequence of real numbers which satisfies an+1≤(1+kn)an−1a_{n+1}\le\left(1+\frac kn\right)a_n-1. Prove that there is an index tt such that at<0a_t<0.
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