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Integer sequence with fibonacci formula

Source: Korea National 2013 #4

November 10, 2013
algebra proposedalgebra

Problem Statement

{an}\{a_n\} is a positive integer sequence such that ai+2=ai+1+ai(i≥1) a_{i+2} = a_{i+1} + a_{i} (i \ge 1) . For positive integer nn, define {bn}\{b_n\} as bn=1a2n+1∑i=14n−2ai b_n = \frac{1}{a_{2n+1}} \sum_{i=1}^{4n-2} { a_i } Prove that bnb_n is positive integer, and find the general form of bnb_n.
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