Question 102
A sequence $\left(a_{n}\right)$ is defined as follows: $a_{1}=\sqrt{2}$, $a_{2}=2$, and $a_{n+1}=a_{n} a_{n-1}^{2}$ for $n \geq 2$. Prove that for every $n \geq 1$ we have
$$
\left(1+a_{1}\right)\left(1+a_{2}\right) \cdots\left(1+a_{n}\right)\lt (2+\sqrt{2}) a_{1} a_{2} \cdots a_{n} .
$$
$$
\left(1+a_{1}\right)\left(1+a_{2}\right) \cdots\left(1+a_{n}\right)\lt (2+\sqrt{2}) a_{1} a_{2} \cdots a_{n} .
$$