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b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i is positive integer

Source: KJMO 2013 p3

May 1, 2019
algebraSequencerecurrence relationSumpositive integerDiophantine equation

Problem Statement

{an}\{a_n\} is a positive integer sequence such that ai+2=ai+1+aia_{i+2} = a_{i+1} +a_i (for all i≥1i \ge 1). For positive integer nn, de fine as bn=1a2n+1Σi=14n−2aib_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i Prove that bnb_n is positive integer.
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