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Prove that a_1 =0 for a sequence of real numbers

Source: Balkan MO ShortList 2007 A8

April 6, 2020

BMOSLalgebraSequencerecurrence relation

Problem Statement

Let

c>2

and

a_0,a_1, \ldots

be a sequence of real numbers such that \begin{align*} a_n = a_{n-1}^2 - a_{n-1} < \frac{1}{\sqrt{cn}} \end{align*} for any

n

\in

\mathbb{N}

. Prove,

a_1=0