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Sequence inequality

Source: China Southeast Math Olympiad 2015 Day 1 P1

June 6, 2017

Sequencesinequalities

Problem Statement

Suppose that the sequence

\{a_n\}

satisfy

a_1=1

and a_{2k}=a_{2k-1}+a_k, a_{2k+1}=a_{2k} for

k=1,2, \ldots

\\Prove that

a_{2^n}< 2^{\frac{n^2}{2}}

for any integer

n \geq 3