Problem

Source: Iran MO 3rd round 2016 mid-terms - Algebra P1

Tags: inequalities, algebra, Sequence, Iran



The sequence $(a_n)$ is defined as: $$a_1=1007$$$$a_{i+1}\geq a_i+1$$Prove the inequality: $$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$