let-U-0-cos-pi-3-and-U-n-1-1-U-n-2-find-U-n-interms-of-n-

Question Number 57407 by Abdo msup. last updated on 03/Apr/19

l e t U_{0} = c o s (\frac{π}{3}) a n d U_{n + 1} = \sqrt{\frac{1 + U_{n}}{2}}

f i n d U_{n} i n t e r m s o f n .

Commented by Abdo msup. last updated on 05/Apr/19

w e h a v e U_{1} = \sqrt{\frac{1 + U_{o}}{2}} = \sqrt{\frac{1 + c o s (\frac{π}{3})}{2}}

= c o s (\frac{π}{2.3}) l e t s u p p o s e U_{n} = c o s (\frac{π}{3 . 2^{n}})

\Rightarrow U_{n + 1} = \sqrt{\frac{1 + U_{n}}{2}} = \sqrt{\frac{1 + c o s (\frac{π}{3 . 2^{n}})}{2}}

= \sqrt{\frac{2 {c o s}^{2} (\frac{π}{3 . 2^{n + 1}})}{2}} = c o s (\frac{π}{3 . 2^{n + 1}}) s o f o r a l l n

U_{n} = c o s (\frac{π}{3 . 2^{n}}) a n d {l i m}_{n \to + \infty} U_{n} = c o s (0) = 1 .