Question

Find the value of the expression $\left(\sqrt{x+\left(\sqrt{x+\left(\sqrt{x....+}\right)}\right)}\right)$.

• A. $\frac{1}{2}[2\sqrt{(2x-1)}+]$
• B. $\frac{1}{2}[\sqrt{(4x+1)}+1]$
• C. $\frac{1}{2}[2\sqrt{(2x-1)}-1]$
• D. $\frac{1}{2}[2\sqrt{(4x-1)}-1]$

Answer 1

The correct option is B $\frac{1}{2}[\sqrt{(4x+1)}+1]$

Let, $y=\sqrt{x+\sqrt{x+\sqrt{x+......}}}$

Squaring both sides we get,

$y^2=x+y$

or, $y^2-y-x=0$

or, $y=\frac{1\pm \sqrt{1+4x}}{2}$

or, $y=\frac{1}{2}[\sqrt{4x+1}+1]$ .... [$y$ can't be negative.]