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-Dedu

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}