Problem

Source: China Northern Mathematical Olympiad 2017

Tags: inequalities, Sequence



A sequence \(\{a_n\}\) is defined as follows: \(a_1 = 1\), \(a_2 = \frac{1}{3}\), and for all \(n \geq 1,\) \(\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}\). Prove that, for all \(n \geq 1\), \(a_1 + a_2 + ... + a_n < \frac{34}{21}\).