Question

The value of $2+\frac{1}{2+\frac{1}{2+...}}$ is

• A. $1_2\sqrt{2}$
• B. $-1-\sqrt{2}$
• C. $1+\sqrt{2}$
• D. $1+2\sqrt{2}$

Answer 1

The correct option is C $1+\sqrt{2}$

Put $x=2+\frac{1}{2+\frac{1}{2+...}}\Rightarrow x=2+\frac{1}{x}\Rightarrow x^2-2x-1=0\Rightarrow x=1+\sqrt{2},1-\sqrt{2}$

$1-\sqrt{2}$ is neglected.
$\therefore x=1+\sqrt{2}\Rightarrow 2+\frac{1}{2+\frac{1}{2+...}}=1+\sqrt{2}$