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a_0 = 1 and a_i^2 > a_{i-1}a_{i+1}, prove a_i > i

Source: Norwegian Mathematical Olympiad 1998 - Abel Competition p1

February 11, 2020

Sequencealgebrainequalities

Problem Statement

Let

a_0,a_1,a_2,...

be an infinite sequence of positive integers such that

a_0 = 1

and

a_i^2 > a_{i-1}a_{i+1}

for all

i > 0

. (a) Prove that

a_i < a_1^i

for all

i > 1

. (b) Prove that

a_i > i

for all

i