U_(n ) is a sequence of real numbers defined by U_0 =0 and for n ∈ N, U_(n+1) =(√(U_n +6)) 1. show that 0≤U_n ≤3. 2. show that U_(n ) is non−decreasing. 3. show that 3−U_(n+1) ≤((3−U_n )/3) 4. Deduct that 0≤3−U


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Question Number 124106 by mathocean1 last updated on 30/Nov/20

$U_{n} i s a s e q u e n c e o f r e a l n u m b e r s$ $d e f i n e d b y U_{0} = 0 a n d f o r n \in N,$ $U_{n + 1} = \sqrt{U_{n} + 6}$ $1 . show that 0 ⩽ U_{n} ⩽ 3 .$ $2 . s h o w t h a t U_{n} i s n o n - d e c r e a s i n g .$ $3 . s h o w t h a t 3 - U_{n + 1} ⩽ \frac{3 - U_{n}}{3}$ $4 . D e d u c t t h a t 0 ⩽ 3 - U_{n + 1} ⩽ {(\frac{1}{3})}^{n}$
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