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convergence of a_(n+1)/a_n

Source: VJIMC 2011 1.2

June 1, 2021
limitsSequences

Problem Statement

Let (an)n=1∞(a_n)^\infty_{n=1} be an unbounded and strictly increasing sequence of positive reals such that the arithmetic mean of any four consecutive terms an,an+1,an+2,an+3a_n,a_{n+1},a_{n+2},a_{n+3} belongs to the same sequence. Prove that the sequence an+1an\frac{a_{n+1}}{a_n} converges and find all possible values of its limit.
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