Question

The value of $\sqrt{7+\sqrt{7-\sqrt{7+\sqrt{7-.....\infty }}}}$ is

• A. $5$
• B. $4$
• C. $3$
• D. $2$

Answer 1

The correct option is C $3$

Let $x=\sqrt{7+\sqrt{7-\sqrt{7+\sqrt{7-........\infty }}}}$
$\Rightarrow \sqrt{7+\sqrt{7-x}}=x$
Squaring both sides, we get
$7+\sqrt{7-x}=x^2$
$\sqrt{7-x}=x^2-7$
Again squaring, we get
$x^4-14x^2+x+42=0$
Option C satisfies above equation.
Hence, the value of the expression is $3$ .