Question 22
A sequence $a_1,a_2,a_3,a_4,\ldots$, with $a_1=\tfrac12$, is defined by
$$a_n=2a_na_{n+1}+3a_{n+1}$$
for all $n=1,2,3,\ldots$. If $b_n=1+1/a_n$ for all $n=1,2,3,\ldots$, find the largest integer $m$ such that
$$\sum_{k=1}^{n}\frac1{\log_3b_k}>\frac m{24}$$
for all positive integer $n\ge2$.
$$a_n=2a_na_{n+1}+3a_{n+1}$$
for all $n=1,2,3,\ldots$. If $b_n=1+1/a_n$ for all $n=1,2,3,\ldots$, find the largest integer $m$ such that
$$\sum_{k=1}^{n}\frac1{\log_3b_k}>\frac m{24}$$
for all positive integer $n\ge2$.