CN-SMO 2006 Problem 4

Given any positive integer $n$ , let $a_n$ be the real root of equation $x^3+\dfrac{x}{n}=1$ . Prove that (1) $a_{n+1}>a_n$ ; (2) $\sum_{i=1}^{n}\frac{1}{(i+1)^2a_i} <a_n$ .