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inequality when a_{n+1} = \sqrt{a_n^2 + 1/a_n }

Source: 1988 Swedish Mathematical Competition p6

March 28, 2021

inequalitiesalgebraSequencerecurrence relationradical

Problem Statement

The sequence

(a_n)

is defined by

a_1 = 1

and

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

for

n \ge 1

. Prove that there exists

a

such that

\frac{1}{2} \le \frac{a_n}{n^a} \le 2

for

n \ge 1