Question

Find the value of $\sqrt{5+\sqrt{5+\sqrt{5}}}+.........\infty$ is equal to

• A. $5$
• B. $5+\sqrt{5}$
• C. $\frac{1+\sqrt{21}}{2}$
• D. $\frac{\sqrt{5}-1}{2}$

Answer 1

The correct option is C $\frac{1+\sqrt{21}}{2}$

Let, $\sqrt{5+\sqrt{5+\sqrt{5}}}+..............=y$
squaring on both side,
$y^2=5+y$
=>$y^2-y-5=0$
=> $y=\frac{1+\sqrt{2}1}{2}$ and $y=\frac{1-\sqrt{2}1}{2}$
but value of square root is always positive,
therefore,
$y=\frac{1+\sqrt{2}1}{2}$