Given that u_(n + 1) = (a_n /2) + 5 evalatuate lim_(x→∞) a_n deduce if a


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Question Number 78156 by Rio Michael last updated on 14/Jan/20

$Given that u_{n + 1} = \frac{a_{n}}{2} + 5$ $e v a l a t u a t e \lim_{x \to \infty} a_{n}$ $d e d u c e i f a_{n} i s c o n v e r g e n t o r d i v e r g e n t .$
Commented by mathmax by abdo last updated on 14/Jan/20

$\Rightarrow 2 u_{n + 1} = u_{n} + 10 \Rightarrow 2 u_{n + 1} - u_{n} - 10 = 0$ $(c e) \to 2 r - 1 = 0 \Rightarrow r = \frac{1}{2} \Rightarrow u_{n} = \frac{λ}{2^{n}} + β$ $u_{0} = λ + β$ $u_{1} = \frac{u_{0}}{2} + 5 = \frac{λ}{2} + β \Rightarrow u_{0} + 10 = λ + 2 β \Rightarrow u_{0} + 10 = λ + 2 (u_{0} - λ) \Rightarrow$ $u_{0} + 10 = - λ + 2 u_{0} \Rightarrow λ = 2 u_{0} - u_{0} - 10 = u_{0} - 10$ $β = u_{0} - λ = u_{0} - u_{0} + 10 = 10 \Rightarrow u_{n} = \frac{u_{0} - 10}{2^{n}} + 10$ $i f u_{0} \neq 10 {l i m}_{n \to + \infty} u_{n} = 10$ $i f u_{0} = 10 \Rightarrow u_{n} = 10 \forall n \Rightarrow l i m u_{n} = 10$
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