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a_{n+1} = (p+1)a_n - pa_{n-1} , a_n \le b_n

Source: Austrian Federal Competition For Advanced Students 2008, Part 1, p3

August 30, 2019
inequalitiesSequencerecurrence relationalgebraAustria

Problem Statement

Let p>1p > 1 be a natural number. Consider the set FpF_p of all non-constant sequences of non-negative integers that satisfy the recursive relation an+1=(p+1)an−pan−1a_{n+1} = (p+1)a_n - pa_{n-1} for all n>0n > 0. Show that there exists a sequence (ana_n) in FpF_p with the property that for every other sequence (bnb_n) in FpF_p, the inequality an≤bna_n \le b_n holds for all nn.
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