I want to prove following statement:
$$F_{n+1}=2+\prod_{k=0}^{n} F_k$$
where $F_n$ is the $n$th Fermat number.
I used proof by induction and got following expression in the induction step:
$$F_{n + 2} = 2 + \prod_{k = 0}^{n + 1} F_k = 2 + F_{n + 1} \prod_{k = 0}^n F_k$$
By using the equation for $F_{n + 1}$, I got $F_{n+2} = 2 + F_{n + 1} (F_{n + 1} - 2)$ and after simplifying I evaluated the following expression:
$$(F_{n + 1} - 1)^2 + 1$$
However I am not able to show how this is equal to $F_{n + 2}$. I know that there is a similar post to this question but it did not help me out at first sight. Could anyone help me out and state what I did wrong or have to do?
