Question

If $x=\sqrt{1+\sqrt{1+\sqrt{1+\ldots \infty}}}$, then $x$ is equal to

(A) $\frac{1+\sqrt{5}}{2}$

(B) $\frac{1-\sqrt{5}}{2}$

(C) $\frac{1 \pm \sqrt{5}}{2}$

(D) None of these

Answer 1

We have

$x=\sqrt{1+\sqrt{1+\sqrt{1+\ldots . \text { to } \infty}}}$

$\Rightarrow  x=\sqrt{1+x} \Rightarrow x^2-x-1=0$

$\Rightarrow  x=\frac{1 \pm \sqrt{1+4}}{2}=\frac{1 \pm \sqrt{5}}{2}$

As $x>0$, we take only $x=\frac{1+\sqrt{5}}{2}$.

So, The correct option will be (A).