Hi I'm studying for my finals and having trouble with this question:

Consider the following recurrence relation:

for where .

Write the corresponding generating function in its closed form.

I've set up the generating function as

and I've gotten as far in my working as:

But according to the solutions I should at this stage be getting:

What am I doing wrong?

I'd really appreciate some help on this one, thanks in advance!

Cheers,

Roro