TITLE: Representing a Decimal as a Fraction - 2 Methods
QUESTION [1 upvotes]: 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.

REPLY [0 votes]: I think you have done fine.
One way of summing a geometric progression is to take:
$$S=\sum_{r=0}^nax^r$$
$$S-xS=a(1-x^{n+1})$$
$$S=a\cdot\frac{1-x^{n+1}}{1-x}$$
And if $-1\lt x\lt 1$, in the limit as $n\to \infty$ $$S=\frac a{1-x}$$
The process you have used echoes this process for the general case. In fact it relies on a repeating decimal being a geometric progression. If you are asked in a question to use the fact that this is a geometric progression, you simply need to make this explicit.
As you can see from the other answers, your method is simpler and less prone to error, and I have always preferred this direct method for use with repeating decimals, because I find it easier to see what is going on (especially when there is a non-repeating bit).
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