Problem

Source: III International Festival of Young Mathematicians Sozopol 2012, Theme for 10-12 grade and Croatian 2nd TST 2012

Tags: algebra, Sequence



The sequence $\{x_n\}_{n=0}^\infty$ is defined by the following equations: $x_n=\sqrt{x_{n-1} x_{n-2}+\frac{n}{2}}$ ,$\forall$ $n\geq 2$, $x_0=x_1=1$. Prove that there exist a real number $a$, such that $an<x_n<an+1$ for each natural number $n$.