What is a good way of summarizing when "generating function" approach is useful to an audience of practitioners?

I'm giving a talk on training neural networks (paper), and want to motivate the idea of generating functions to audience not exposed to this concept.

One example I was considering was this: consider following matrix sequences

$$ \begin{array} \ A^0,A^1,A^2,\ldots\\ B^0,B^1,B^2\ldots\\ \end{array} $$

We can consider a new sequence $(A+B)^n$ $$\ (A+B)^0,(A+B)^1,(A+B)^2,\ldots$$

If original sequences had generating functions $G_A, G_B$, combined sequence generating function is this:

$$G_{A+B}=(I-x G_A B)^{-1} G_A$$

In this case, "generating function" approach replaces matrix powering with inversion, so it helps when inversion is easy (ie, $A+B$ is diagonal + rank1), and doesn't help if matrix powering is easy.