I have that problem to solve and I don't find a correct way to start with.
The probability generating function of a random variable N is $Q_N(z) = 1/6z^7 + 1/3z^2 + 1$. Identify the p.m.f. of N and the variance of N.
I know that to compute a PGF it is the sum of $p_kz^k$ up to k. I know how to compute the MGF but I don't understand how I could to the process backwards
Assuming that $X$ is a discrete random variable taking values from 0 to n then, from the definition of probability generating function (PGF), $G(z)=E(z^X)$,
$$G(z)=E(z^X)=\sum_k^n z^k P(X=k)=P(X=0)+P(X=1)z+\dots +P(X=n)z^n$$
Thus the problem is to examine the terms which have the same exponents in the definition of $G(z)$ and the given PGF. e.g. for k=7, $G(z)$ contains $P(X=7)z^7$ which is given $\frac{z^7}{6}$ in the question. Thus $P(X=7)=\frac{1}{6}$. Since in the given PGF there is no $z^3$ it simply means that $P(X=3)=0$.
By examining the other terms you can find the PMF of the random variable asked in the question.