I'm working on some proofs on a recurrence relation with Fermat numbers.
a). Prove with Fermat numbers $f_n = 2^{2^n} + 1$ that:
$$ f_0 \cdot f_1 \cdot f_2 \cdot f_3 \cdots f_{n-1} = f_n - 2$$
b). Prove that $f_n$ is relatively prime with $f_0, f_1, f_2, ... , f_{n-1}$. In other words:
for any $n$, $f_n$ is not divisible with any factors of its previous Fermat number.
In my attempt on (a) I used the sum of the powers of $2$ :
$$2^0 + 2^1 + 2^2 + 2^3 + ... + 2^n = 2^{n+1} - 1$$
but I didn't come very far.
Some help would be very much appreciated.