{ ((u_(n+1) = u_n −3 )),((v_(n+1) = 4v_n )) :} : u_0 = v_0 = 1 w_n = ((1−u_n )/v_n ) − show that w_n is bounded − find a,b∈R such that a ≤ w


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Question Number 190464 by alcohol last updated on 03/Apr/23
${ ((u_(n+1) = u_n −3 )),((v_(n+1) = 4v_n )) :} : u_0 = v_0 = 1 w_n = ((1−u_n )/v_n ) − show that w_n is bounded − find a,b∈R such that a ≤ w_n ≤ b$
${\begin{cases} u_{n + 1} = u_{n} - 3 \\ v_{n + 1} = 4 v_{n} \end{cases} : u_{0} = v_{0} = 1$ $w_{n} = \frac{1 - u_{n}}{v_{n}}$ $- s h o w t h a t w_{n} i s b o u n d e d$ $- f i n d a, b \in R s u c h t h a t a ⩽ w_{n} ⩽ b$
	Answered by mr W last updated on 03/Apr/23

	$u_{0} = 1$ $u_{1} = 1 - 3$ $u_{2} = 1 - 3 \times 2$ $. . .$ $u_{n} = 1 - 3 \times n = 1 - 3 n$ $v_{0} = 1$ $v_{1} = 1 \times 4$ $v_{2} = 1 \times 4^{2}$ $. . .$ $v_{n} = 1 \times 4^{n} = 4^{n}$ $w_{n} = \frac{1 - (1 - 3 n)}{4^{n}} = \frac{3 n}{4^{n}}$ $\lim_{n \to \infty} w_{n} = 0$ $0 ⩽ w_{n} ⩽ \frac{3}{4}$ $f o r a ⩽ w_{n} ⩽ b :$ $a = 0, b = \frac{3}{4}$
	Commented by alcohol last updated on 03/Apr/23

	$a l r i g h t t h a n k y o u$ $w h a t a b o u t t h e b o u n d s ?$
	Commented by mr W last updated on 03/Apr/23

	$i m i s r e a d . w_{n} = \frac{3 n}{4^{n}} .$
	Commented by alcohol last updated on 03/Apr/23

	$t h a n k y o u$ $w_{n} = \frac{3 n}{4^{n}} r i g h t ?$ $p l e a s e h o w d i d y o u g e t {(\frac{3}{4})}^{n} ?$
	Commented by mr W last updated on 03/Apr/23

	$w_{0} = 0 = m i n$ $w_{1} = \frac{3}{4} = m a x$ $0 ⩽ w_{n} ⩽ \frac{3}{4}$
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