Question

Find the value of $\sqrt{2+\sqrt{2+\sqrt{2}+.......\infty }}$

Answer 1

Let $x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...\infty }}}}$

On squaring both sides, we get

$x^2=2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...\infty }}}}$

$x^2=2+x$

$x^2-x-2=0$

$x=\frac{1_{-}^{+}\sqrt{1^2-4\ast 1\ast -2}}{2\left(1\right)}$

$x=\frac{1_{-}^{+}\sqrt{9}}{2}==\frac{1_{-}^{+}3}{2}$

$x=2$ or $x=-1$

Since a negative number cannot be under root, hence $x=2$.