I have the following problem:
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?
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.