<s>[INST] Find the base system, $x$, such that $\frac{1}{5}$ and $\overline{.17}$ are numerals for the same number. So far I know that $\frac{1}{5} = 0.2$ in base 10. However, I'm not really sure how to convert it into another base. Any suggestions would be greatly appreciated. 
 [/INST] $$\begin{align} (0.1717...)_b =& \left(\frac{1}{b}+\frac{7}{b^2}\right) +
\left(\frac{1}{b^3}+\frac{7}{b^4}\right) + \cdots \\
=& \frac{b+7}{b^2}+\frac{b+7}{b^4} +\cdots \\
=& \sum_{n=1}^{\infty} \frac{b+7}{b^{2n}} \\
=& (b+7)\sum_{n=1}^{\infty}(b^{-2})^n \\
=& \frac{b+7}{b^2-1}
\end{align}$$
and this is equal to $1/5$, so $b=9$.
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