Evaluate this infinite product involving $a_k$

Question

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute:

$$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$

Off the bat, we can seperate $a_0$

$$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - \frac{1}{a_k}$$

Lets see:

$a_0 = 5/2$

$a_1 = 25/4 - 2 = 17/4$

$a_2 = 289/16 - 2 = 257/16$

$$P = (-3/2)\cdot(-13/4)\cdot(-241/16).....$$

Lets compute the first three for $P_3$

$P_1 = -3/2 = (-192/128)$

$P_2 = (-3/2)(-13/4) = (39/8) = (624/128)$

$P_3 = (39/8)(-241/16) = (-9399/128)$

But it is difficult to find a pattern for $P_k$

Help?

Thanks =)

Answer 1

$\textbf{Hint:}$

show this by induction on $n\ge0$ $$a_n=2^{2^n}+2^{-2^n},\quad n=0,1,2\dots$$

and notice that $a_k+1=a_{k-1}^2-1=(a_{k-1}-1)(a_{k-1}+1),$ $$1-\frac{1}{a_k}=\frac{a_k-1}{a_k}=\frac{a_{k+1}+1}{a_k+1}\cdot\frac{1}{a_k}$$