Prove that the mean of the $u_i$'s generated by a congruential linear generator of complete period is $\frac{1}{2} -\frac{1}{2m}$

Question

Congruential linear generator (pseudo random numbers)
I have to prove this proposition but I don't know how. I tried by induction on $m$, the modulus of the generator, but it hadn't got me anywhere:

Prove that the mean of the $u_i$'s generated by a congruential linear generator of complete period is $$\frac{1}{2} -\frac{1}{2m}$$ where every $u_i = \frac{z_i}{m}$ and $z_i = a^iz_0 + c\frac{(a^i - 1)}{a-1}\mod m$.

This theorem says when is a clg of complete period:

A clg has complete period if and only if the following three conditions are satisfied:

1. $c$ and $m$ are relative primes.
2. If $q$ is a prime that divides $m$, then $q$ also divides $a-1$.
3. If $4$ divides $m$, $4$ divides $a-1$."

Any idea can help me, I'm really lost. Thanks.