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Regional Olympiad - FBH 2016 Grade 12 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

September 22, 2018

limitalgebraSequence

Problem Statement

Let

a_1=1

and

a_{n+1}=a_{n}+\frac{1}{2a_n}

for

n \geq 1

. Prove that

a)

n \leq a_n^2 < n + \sqrt[3]{n}

b)

\lim_{n\to\infty} (a_n-\sqrt{n})=0