We can use induction to prove this. We already know the first five fermat numbers:
$$\begin{gather} F_0 = 3 \\ F_1 = 5 \\ F_2= 17 \\ F_3 = 257 \end{gather}$$
The statement we are trying to prove is:
$$ F_0 F_1 F_2 \cdots F_{n-1} + 2 = F_n$$
We know that this statement is true when \(n \le 3\). This means if \(n \le 3\):
$$\begin{align} F_0 F_1 F_2 \cdots F_{n-1} F_n &= (F_0 F_1 F_2 \cdots F_{n-1}) F_n \\ &= (F_n - 2) F_n \\ &= (2^{2^n} -1)(2^{2^n}+1) \\ &= 2^{(2* 2^n)} - 1 = 2^{2^{n+1}} + 1 -2 \\ &= F_{n+1} - 2 \end{align}$$
This shows if the statement if true for \(n\), then it would be true for \(n+1\).