Question

$\sqrt{x + 2 \sqrt{x + 2 \sqrt{x + 2 \sqrt{3x } } } } = x$
answer this please

Answer 1

Here,  from right to left, $\sqrt{x + 2}$ is repeating 3 times (or it happens endlessly also!). 

So in such problems, as we do from right to left, same answer is repeating as each $\sqrt{x + 2}$ is being passed through. 

∴ $\sqrt{x + 2\sqrt{x + 2\sqrt{x + 2\sqrt{3x}}}} = \sqrt{x + 2\sqrt{x + 2\sqrt{3x}}} = \sqrt{x + 2\sqrt{3x}}$
$= \sqrt{3x} = x$

$3x = x^2 \\ = 3 * x = x * x$

∴ $x = 3$. 

Let's check. 

$\sqrt{3 + 2\sqrt{3 + 2\sqrt{3 + 2\sqrt{3 * 3}}}} \\ = \sqrt{3 + 2\sqrt{3 + 2\sqrt{3 + 2 * 3}}} \\ = \sqrt{3 + 2\sqrt{3 + 2\sqrt{3 + 6}}} \\ = \sqrt{3 + 2\sqrt{3 + 2\sqrt{9}}} \\ = \sqrt{3 + 2\sqrt{3 + 2 * 3}} \\ = \sqrt{3 + 2\sqrt{3 + 6}} \\ = \sqrt{3 + 2\sqrt{9}} \\ = \sqrt{3 + 2 * 3} \\ = \sqrt{3 + 6} \\ = \sqrt{9} \\ = 3$

If $\sqrt{x + (n - 1)\sqrt{x + (n - 1)\sqrt{x + (n - 1).........\sqrt{x + (n - 1)\sqrt{nx}}}}} = x$, then $x = n$. 

Hope this will be helpful.