Problem
Find the number that represented by $[2,2,2 \ldots]$
I know it wasn't difficult, but I was absent the last two classes. So I just want to make sure that I got it right.
My attempt was,
Consider $x = [2;\overline{2}] \implies [2;x]$ Hence, $x = 2 + \dfrac{1}{x} \Leftrightarrow x = \dfrac{2x + 1}{x} \Leftrightarrow x^2 - 2x - 1 = 0$. Solving the quadratic equation we have $x = \sqrt{2} + 1$ since $x > 0$. Therefore $[2,2,2, \ldots] = \sqrt{2} + 1$
Is it correct?
Thank you,