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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!

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  • $\begingroup$ Hint: If you wee to substitute $z=x^i$, what would your GF look like then? $\endgroup$ – awkward Mar 25 '19 at 23:26
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Hint: make the substitution $u=x^i$.

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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

$$1 + x^i + x^{2i} + x^{3i} + ... + x^{ki} = 1 + (x^i)^1 + (x^i)^2 + (x^i)^3 + ... + (x^i)^k = \frac{1 - (x^i)^{k+1}}{1 - x^i}$$

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  • $\begingroup$ Thank you so much. That makes sense, it's just a simple substitution! $\endgroup$ – Jeffery Rice Mar 25 '19 at 23:38

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