let-u-0-3-and-u-n-1-2-u-n-2-calculate-u-n-interms-of-n-

Question Number 32293 by abdo imad last updated on 22/Mar/18

l e t u_{0} = \sqrt{3} a n d u_{n + 1} = \sqrt{2 + u_{n}^{2}}

c a l c u l a t e u_{n} i n t e r m s o f n .

Commented by abdo imad last updated on 24/Mar/18

l e t p u t v_{n} = u_{n}^{2} w e h a v e v_{n + 1} = u_{n + 1}^{2} = 2 + u_{n}^{2} 2 + v_{n} s o

(v_{n}) i s a r i t h m e t i c s e q u e n c e \Rightarrow v_{n} = v_{0} + n r = v_{0} + 2 n b u t

v_{0} = u_{0}^{2} = 3 \Rightarrow v_{n} = 2 n + 3 d u e t o u_{n} > 0 w e h a v e

u_{n} = \sqrt{v_{n}} = \sqrt{2 n + 3} .

Answered by mrW2 last updated on 24/Mar/18

u_{n}^{2} - u_{n - 1}^{2} = 2

\dots \dots

u_{2}^{2} - u_{1}^{2} = 2

u_{1}^{2} - u_{0}^{2} = 2

\Rightarrow u_{n}^{2} - u_{0}^{2} = 2 n

\Rightarrow u_{n}^{2} = u_{0}^{2} + 2 n = 2 n + 3

\Rightarrow u_{n} = \sqrt{2 n + 3}

Commented by abdo imad last updated on 24/Mar/18

y o u r a n s w e r i s c o r r e c t s i r t h a n k s \dots