From what I understand, there cannot be a generating function for the series $1, 2, 1, 1, 1, 1$, and I have not managed to find any examples of similar generating functions. However, I am expected to find one.
How can it be possible to modify the generating function $(\frac{x} {(1-x)})$such that only the n=1st term is increased by $1$? It does not seem possible.
Just use $$\frac{x}{1-x}+x^2$$