A Quadratic Map and a Sum

Problem

A sequence has $a_1 = \dfrac52$ and $2a_{n+1} = a_n^2 + 1$ . Find the largest integer $m$ such that

\frac{2a_1 - 1}{a_1 + 1} + \frac{2a_2 - 1}{a_2 + 1} + \cdots + \frac{2a_{2018} - 1}{a_{2018} + 1} > m.

Answer

Solution

Difficulty	8/10
Topics	sequences, Estimation, Telescoping

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