Representing a Decimal as a Fraction - 2 Methods So I am trying to represent the number 0.71717171 · · · as a fraction and have managed to do it using algebra. I was told I was supposed to solve it using a geometric sum.
Could someone guide me through the steps of solving such a question with geometric sums?
My method:
Let $x = 0.\overline{71}$
Then $100x = 71.\overline{71}$
Consequently, $$100x - x = 71.\overline{71} - 0.\overline{71}$$
$$99x = 71$$
$$x=\frac{71}{99}$$
Any help would be greatly appreciated. Thank you.

Using geometric sums:

$$x=\dfrac{7}{10}+\dfrac{1}{10^2}+\dfrac{7}{10^3}+\dfrac{1}{10^4}+\dfrac{7}{10^5}+\ldots$$
  $$x= \left( \dfrac{7}{10}+\dfrac{7}{10^3}+\dfrac{7}{10^5}+\ldots \right)+\left( \dfrac{1}{10^2}+\dfrac{1}{10^4}+\dfrac{1}{10^6}+\ldots \right) $$
  Here you have a decreasing geometric  with ratio $r<1$, so:
  $$x=\left( \dfrac{\dfrac{7}{10}}{1-\dfrac{1}{10^2}} \right)+ \left( \dfrac{\dfrac{1}{10^2}}{1-\dfrac{1}{10^2}} \right)$$
  $$x=\frac{71}{99}$$

