let U_0 =cos((π/3)) and U_(n+1) =(√((1+U_n )/2)) find U


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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

$w e h a v e U_{1} = \sqrt{\frac{1 + U_{o}}{2}} = \sqrt{\frac{1 + c o s (\frac{π}{3})}{2}}$ $= c o s (\frac{π}{2.3}) l e t s u p p o s e U_{n} = c o s (\frac{π}{3 . 2^{n}})$ $\Rightarrow U_{n + 1} = \sqrt{\frac{1 + U_{n}}{2}} = \sqrt{\frac{1 + c o s (\frac{π}{3 . 2^{n}})}{2}}$ $= \sqrt{\frac{2 {c o s}^{2} (\frac{π}{3 . 2^{n + 1}})}{2}} = c o s (\frac{π}{3 . 2^{n + 1}}) s o f o r a l l n$ $U_{n} = c o s (\frac{π}{3 . 2^{n}}) a n d {l i m}_{n \to + \infty} U_{n} = c o s (0) = 1 .$
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