u-n-1-2-u-n-show-that-u-n-1-u-n-and-u-n-u-n-1-have-same-sign-

Question Number 166648 by alcohol last updated on 24/Feb/22

u_{n + 1} = \sqrt{2 + u_{n}}

s h o w t h a t u_{n + 1} - u_{n} a n d u_{n} - u_{n - 1}

h a v e s a m e s i g n

Answered by mr W last updated on 24/Feb/22

u_{n + 1} = \sqrt{u_{n} + 2} ⩾ 0

a t m o s t o n e t e r m c a n b e z e r o, a l l o t h e r

t e r m s a r e p o s i t i v e, i . e . u_{n + 1} + u_{n} > 0 .

u_{n + 1}^{2} = u_{n} + 2

u_{n}^{2} = u_{n - 1} + 2

u_{n + 1}^{2} - u_{n}^{2} = u_{n} - u_{n - 1}

(u_{n + 1} - u_{n}) (u_{n + 1} + u_{n}) = u_{n} - u_{n - 1}

\frac{u_{n} - u_{n - 1}}{u_{n + 1} - u_{n}} = u_{n + 1} + u_{n} > 0

\Rightarrow u_{n + 1} - u_{n} a n d u_{n} - u_{n - 1} h a v e s a m e s i g n .

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