Average of the numbers given by a linear congruential generator

Question

Show that the average of numbers given by a full cicle of a linear congruential generator with full lenght period is $\frac{1}{2}-\frac{1}{2m}$, whith $m$ as the module of the generator.

I began by using the definition of a linear congruential generator and the Hull & Dobell theorem, but I got lost in a cycling argument.

Answer 1

Hint: if the generator has full period, each possible value must show up once. What is the average of those?