Find the base system, $x$, such that $\frac{1}{5}$ and $\overline{.17}$ are numerals for the same number.

Question

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.

Answer 1

$$(x^2-1)\cdot\dfrac15=(17.17\cdots)_x-(.17\cdots)_x=(17)_x=1\cdot x^1+7$$

Answer 2

$0.171717..._x=\dfrac{17_x}{x^2}+\dfrac{17_x}{x^4}+\dfrac{17_x}{x^6}+...$

$=\dfrac{17_x/x^2}{1-1/x^2}=\dfrac{17_x}{x^2-1}=\dfrac{x+7}{x^2-1}=\dfrac15$

Answer 3

$$\begin{align} (0.1717...)_b =& \left(\frac{1}{b}+\frac{7}{b^2}\right) + \left(\frac{1}{b^3}+\frac{7}{b^4}\right) + \cdots \\ =& \frac{b+7}{b^2}+\frac{b+7}{b^4} +\cdots \\ =& \sum_{n=1}^{\infty} \frac{b+7}{b^{2n}} \\ =& (b+7)\sum_{n=1}^{\infty}(b^{-2})^n \\ =& \frac{b+7}{b^2-1} \end{align}$$ and this is equal to $1/5$, so $b=9$.