I need help making an OGF for $1 + x^i + x^{2i}+...+x^{ki}$. I already know how to verify that $1 +x +x^2+...+x^k$ can be written by $({1-x^{k+1}})/({1-x})$. I'm wondering if there is any correlation between the two..?
Any help would be greatly appreciated. Thanks!
On
Notice that
$$1 + x^i + x^{2i} + x^{3i} + ... + x^{ki} = 1 + (x^i)^1 + (x^i)^2 + (x^i)^3 + ... + (x^i)^k$$
This is a finite geometric series with ratio $x^i$, and thus
$$1 + x^i + x^{2i} + x^{3i} + ... + x^{ki} = 1 + (x^i)^1 + (x^i)^2 + (x^i)^3 + ... + (x^i)^k = \frac{1 - (x^i)^{k+1}}{1 - x^i}$$
Hint: make the substitution $u=x^i$.