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Here's my work:

[tex]x = 1 + \frac{1}{2+x} \Rightarrow x^2 + x - 3 = 0[/tex] and by quadratic formula, we get [tex]x = \frac{-1 \pm \sqrt{13}}{2}[/tex] but we only want the positive root so I get [tex]x = \frac{-1 + \sqrt{13}}{2}[/tex] for my answer but the answer given was [tex]x = \frac{1 + \sqrt{3}}{2}[/tex], so I'm confused at which it is....

Moreover, I can't seem to find any other example except for [tex]\langle 1, 1, 1, \ldots \rangle[/tex] to see if I'm doing my computation right. Please help.