Question

Find the value of $\sqrt{6+\sqrt{6+\sqrt{6+\dots \dots .}}}$

Answer 1

Let $x=\sqrt{6+\sqrt{6+\sqrt{6+\cdots }}}$

$⟹x=\sqrt{6+x};x>0$

Squaring on both sides

$⟹x^2=6+x$

$⟹x^2-x-6=0$

$⟹x^2-3x+2x-6=0$

$⟹(x-3)(x+2)=0$

$⟹x=3,-2$

$⟹$ $\sqrt{6+\sqrt{6+\sqrt{6+\cdots }}}=3$