# Lots of logarithms

Algebra Level pending

A sequence $${a_n}$$ is defined as follows:

$$a_1=3, a_{k+1}=4a_k+3$$

If $$\sum_{k=1} ^{1000} \frac{1}{\log_2(a_k+1)\log_2(a_{k+1}+1)}$$ can be expressed as $$\frac{m}{n}$$ for positive integers $$m$$ and $$n$$, find $$m+n$$.

×