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If $x = 1 + \frac{x}{1 + \frac{x}{1+ \frac{x}{1 + \cdots}}}$, then what is $x$?

Jun 25, 2019

#1
+91
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So, when translating to LaTeX, we get $$x = 1 + \frac{x}{1 + \frac{x}{1+ \frac{x}{1 + \cdots}}}$$

Look at all the fractions under the first x. That is $$1 + \frac{x}{1 + \frac{x}{1+ \frac{x}{1 + \cdots}}}$$

But that is also x!

So, x = 1 + x/x which is x = 1 + 1 which is 2.

Jun 25, 2019

#1
+91
+2

So, when translating to LaTeX, we get $$x = 1 + \frac{x}{1 + \frac{x}{1+ \frac{x}{1 + \cdots}}}$$

Look at all the fractions under the first x. That is $$1 + \frac{x}{1 + \frac{x}{1+ \frac{x}{1 + \cdots}}}$$

But that is also x!

So, x = 1 + x/x which is x = 1 + 1 which is 2.

Pushy Jun 25, 2019