A) \( \Large \frac{1+\sqrt{5}}{2} \) |
B) \( \Large \frac{1-\sqrt{5}}{2} \) |
C) \( \Large \frac{1\pm \sqrt{5}}{2} \) |
D) none of these . |
A) \( \Large \frac{1+\sqrt{5}}{2} \) |
We have, \( \Large x=\sqrt{1+\sqrt{1+\sqrt{1+.....\infty}}} \)
=> \( \Large x=\sqrt{1+x} \)
=>\( \Large x^{2}=1+x => x^{2}-x-1=0 \)
=> \( \Large x=\frac{1\pm \sqrt{1+4}}{2}=\frac{1\pm \sqrt{5}}{2} \)
As x > 0 we take only \( \Large x=\frac{1+\sqrt{5}}{2} \)