Question 42
A sequence $a_{1}, a_{2}, a_{3}, \ldots$ with initial terms $a_{1}=4$ and $a_{2}=16$ is given, for which $\log _{2}\left(\log _{2} a_{1}\right), \log _{2}\left(\log _{2} a_{2}\right), \log _{2}\left(\log _{2} a_{3}\right), \ldots$ is an arithmetic sequence. Prove that
$$
\log _{2}\left(\log _{2}\left(4 a_{1} a_{2} \ldots a_{n}\right)\right)=n+1
$$
for all natural numbers $n$.
(20 points)
$$
\log _{2}\left(\log _{2}\left(4 a_{1} a_{2} \ldots a_{n}\right)\right)=n+1
$$
for all natural numbers $n$.
(20 points)