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I have the following problem: enter image description here

The way I solved the problem is by rewriting it as a system of equations:

$$ x = 1 +\frac 1y$$ $$ y = 2 + \frac1y$$ Then I solved it as a normal system of equations and arrived at the answer of $\sqrt2$. But apparently the answer is $e$? How does that make sense? Can someone explain the result to me?

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    $\begingroup$ Can you show us how you solved it? $\endgroup$ – Randall Jul 22 at 18:17
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    $\begingroup$ This is equal to $\sqrt{2}$. I don't know where you got this equals $e$ from. $\endgroup$ – Peter Foreman Jul 22 at 18:22
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    $\begingroup$ Continued fraction for $e$ is not periodic $\endgroup$ – J. W. Tanner Jul 22 at 18:28
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    $\begingroup$ @Adam Grey:$\;$Out of curiosity, what book? $\endgroup$ – quasi Jul 22 at 18:28
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    $\begingroup$ @Adam Grey: To solve without a system, consider the continued fraction for $x+1$. Then immediately, you get the equation $$x+1=2+\frac{1}{x+1}$$ which yields $x=\sqrt{2}$. $\endgroup$ – quasi Jul 22 at 18:33
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As far as I know, $e$ is not so good looking at continued fractions... Indeed $$e=[2;1,2,1,1,4,1,1,6,1,1,8,\ldots]$$You have found a correct answer, yes! $x=\sqrt{2}$ is the correct number with the given continued fraction.

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  • $\begingroup$ @TonyK Thanks for noting, it was a small typo... $\endgroup$ – Anand Jul 22 at 18:46
  • $\begingroup$ Do you know that \ldots and \dots are rendered in the same way. $\endgroup$ – Peter Foreman Jul 22 at 19:20

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