How to create a generating function that only includes multiples of 4?

Question

I am going through generating functions to solve for the number of unordered selections or distributions. My text book asks the question:

Give a formula similar to (1) for $1 + x^4 + x^8 + ... + x^{24} $

(1) is given as $\frac{1-x^{m +1}}{1-x}$ = $1 + x + x^2 + ...+ x^m$

So I see that the functions are similar and instead of the exponent increasing by 1 each element, we instead increase by multiples of 4. But how can we manipulate (1) so that it resembles the given equation?

Answer 1

Let's write $x^4 = t$ and the required equation just becomes $1 + t + t^2 + ... + t^{m} $

and is given as $\frac{1-t^{m +1}}{1-t}$

Answer 2

$s=1+x^4+x^8+...+x^{24}\\x^4s=x^4+x^8+...+x^24+x^{28}\\ compute- (s-x^4s)\\s-x^4s=1-x^{28}\\s(1-x^4)=1-x^{28}\\s=\frac{1-x^{28}}{1-x^{4}}$

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