<s>[INST] Let $\frac{{{x^5}}}{{1 - {x^5} + {x^{10}}}} = k$. What is $\frac{{{x^{15}}}}{{1 - {x^{15}} + {x^{30}}}}$? Let $\frac{{{x^5}}}{{1 - {x^5} + {x^{10}}}} = k$ and $k\in \mathbb{R}$.
What is $\frac{{{x^{15}}}}{{1 - {x^{15}} + {x^{30}}}}$?
 [/INST] Note that
$$
k=\frac{x^5}{1-x^5+x^{10}}=\frac{1}{x^{-5}-1+x^{5}}\implies x^{-5}-1+x^{-5}=k^{-1}.
$$
It follows that
\begin{aligned}
k^{-2}&=1+(x^{-10}+x^{10})-2(x^{-5}-1+x^5)\implies(x^{-10}+x^{10})=k^{-2}-1+2k^{-1},\\
k^{-3}&=(x^{-5}-1+x^{-5})^3=(x^{-15}-1+x^{15})+6(x^{-5}-1+x^5)-3(x^{-10}+x^{10}).
\end{aligned}
We conclude
$$
x^{-15}-1+x^{-15}=k^{-3}-6k^{-1}+3(k^{-2}-1+2k^{-1})=k^{-3}+3k^{-2}-3
$$
and the desired answer the reciprocal of the rightmost expression above:
$$
\frac{k^3}{1+3k-3k^3}.
$$
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