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Solve the equation

$$x=2+\dfrac1{2+\dfrac1{...2+\dfrac1{2+\dfrac1x}}}$$

Where there are n layers in the fraction

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  • $\begingroup$ you can accept my answer if it was good, if you want more details ask :D $\endgroup$ – user58512 Mar 12 '13 at 12:01
  • $\begingroup$ Nope I understood it after you explained. Thanks allow for your help :) $\endgroup$ – user61067 Mar 12 '13 at 15:53
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the first thing to observe is that the number of layers doesn't matter. If $x = 2 + \frac{1}{x}$ it solves your equation, and simple continued fractions have a unique assigned value.

It's easy to solve $x = 2 + \frac{1}{x}$ though, just subtract two then multiply up to get $x^2-2x - 1 = 0$.

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  • $\begingroup$ Did you mean $x=2+\frac{1}{x}$? $\endgroup$ – Vincent Tjeng Mar 3 '13 at 17:28
  • $\begingroup$ @VincentTjeng, yes thank you. corrected now. $\endgroup$ – user58512 Mar 3 '13 at 17:30
  • $\begingroup$ @user58512 I can see that the number of layers doesn't matter, but can you tell me the reasoning behind this? $\endgroup$ – user61067 Mar 5 '13 at 11:54
  • $\begingroup$ @user61067, I did! If you solve the single layered one you solve all lays.. and there's a unique solution. $\endgroup$ – user58512 Mar 5 '13 at 12:45
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    $\begingroup$ Ah perfect! Thank you!! $\endgroup$ – user61067 Mar 5 '13 at 15:53

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