How to multiply generating functions with $x^n$ and $x^{5n}$ and $x^{2n}$

Question

I have to solve a combinatoric problem using generating functions, how to further simplify this? $$\sum_{n=0}^\infty {x^n}*\sum_{n=0}^\infty {x^{5n}}*\sum_{n=0}^\infty x^{2n}$$

Answer 1

These are all geometric series and can be summed that way. $$\sum_{n=0}^\infty x^n=\frac 1{1-x}\\\sum_{n=0}^\infty x^{5n}=\sum_{n=0}^\infty (x^5)^n$$

Answer 2

The product is equal to $$ \frac{1}{(1-x)(1-x^2)(1-x^5)} $$ by using the geometric series.