let-u-0-u-1-1-and-u-n-1-u-n-u-n-1-2-find-u-n-3-let-x-0-the-positif-roots-of-x-2-x-1-4-prove-that-n-2-x-0-n-2-u-n-x-0-n-1-

Question Number 43541 by abdo.msup.com last updated on 11/Sep/18

l e t u_{0} = u_{1} = 1 a n d u_{n + 1} = u_{n} + u_{n - 1}

2) f i n d u_{n}

3) l e t x_{0} t h e p o s i t i f r o o t s o f x^{2} = x + 1

4) p r o v e t h a t \forall n ⩾ 2 x_{0}^{n - 2} ⩽ u_{n} ⩽ x_{0}^{n - 1} .

Answered by behi83417@gmail.com last updated on 12/Sep/18

\frac{u_{n + 1}}{u_{n}} = 1 + \frac{u_{n - 1}}{u_{n}} \Rightarrow x = 1 + \frac{1}{x}

\Rightarrow x^{2} - x - 1 = 0 \Rightarrow x = \frac{1 + \sqrt{5}}{2}

\Rightarrow \frac{u_{n + 1}}{u_{n}} = \frac{\sqrt{5} + 1}{2} \dots .