Question

$\sqrt{2+\sqrt{2+\sqrt{\mathrm{2..........}\infty }}}$ = $x$ . Find $x$

Answer 1

Given $\sqrt{2+\sqrt{2+\sqrt{2+\cdots \infty }}}=x$

Squaring on both sides

$⟹2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots \infty }}}=x^2$

$⟹2+x=x^2$

$⟹x^2-x-2=0$

$⟹x^2-2x+x-2=0$

$⟹(x-2)(x+1)=0$

$⟹x=2,-1$

As $x>0$ so $x=2$