u_(n+1) = (√(2+u_n )) show that u_(n+1) −u_n and u_n −u


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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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