Question

The value of $\sqrt{2+\sqrt{2+\sqrt{\mathrm{2............}}}}$ is :

Answer 1

Consider the given expression,

Let,

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

$x=\sqrt{2+x}$

$x^2=2+x$

$x^2-x-2=0$

$x^2-2x+x-2=0$

$x(x-2)+1(x-2)=0$

$(x-2)(x+1)=0$

$x=2orx=-1$

Hence, this is the answer.