Any hint for this product of complex numbers I have to calculate the modulus of $z$. I've already tried to find a general formula for $\left(\frac{1+i}{2}\right)^{2^n}$, which seems to be $\frac{1}{2^{2^{n-1}}}, \forall \geq3$, using the trigonometric form and de Moivre's formula.
$z=\left[1+\frac{1+i}{2}\right] \left[1+\left(\frac{1+i}{2}\right)^2\right] \left[1+\left(\frac{1+i}{2}\right)^4\right] \cdots \left[1+\left(\frac{1+i}{2}\right)^{2^n}\right]$
How should I keep solving this?
Thanks!

Note that 
$$z=
\frac {[1-(\frac {1+i}{2})]\left[1+\frac{1+i}{2}\right] \left[1+\left(\frac{1+i}{2}\right)^2\right] \left[1+\left(\frac{1+i}{2}\right)^4\right] \cdots \left[1+\left(\frac{1+i}{2}\right)^{2^n}\right]        }{[1-(\frac {1+i}{2})]}=$$
$$\frac { \left[1-\left(\frac{1+i}{2}\right)^{2^{n+1}}\right]        }{[1-(\frac {1+i}{2})]}$$
We may simplify it with $$(\frac{1+i}{2})^2 = \frac {i}{2}$$
