Find-the-real-number-satisfying-x-1-1-1-x-

Question Number 184873 by topgrace100 last updated on 13/Jan/23

Find the real number satisfying

x = \sqrt{1 + \sqrt{1 + \sqrt{1 + x}}}

Answered by Rasheed.Sindhi last updated on 13/Jan/23

x = \sqrt{1 + \sqrt{1 + \sqrt{1 + x}}}

x = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \dots}}}}}

x^{2} = 1 + x

x^{2} - x - 1 = 0

x = \frac{1 + \sqrt{1 + 4}}{2} = \frac{1 + \sqrt{5}}{2} ✓ [∵ \frac{1 - \sqrt{5}}{2} < 0]

Commented by Frix last updated on 13/Jan/23

x = \sqrt{\dots} ⩾ 0 \Rightarrow x \neq \frac{1 - \sqrt{5}}{2}

Commented by Rasheed.Sindhi last updated on 13/Jan/23

R i g h t s i r! I^{'} m g o i n g t o g e t o u t \frac{1 - \sqrt{5}}{2}

f r o m t h e s o l u t i o n s e t . T h a n X!