Recursive Sequence $2a_{n+1}=a_n^2+1$

Question

In sequence $\{a_n\}$, $a_1=2$ and $2a_{n+1}=a_n^2+1$, define $b_n=\cfrac{2a_n-1}{a_n+1}$, if $$b_1+b_2+\cdots+b_{2019}>t$$find the max integer t.

I find it is nearly impossible to find the general form of $a_n$ or $b_n$, but how do we solve this?

Answer 1

Hint: $a_n \to \infty$ monotonically, $b_n = 2 - \dfrac{3}{a_n+1}$. This implies $b_1 + \cdots + b_n \approx 2n-3$ for large $n$.