a-n-2-a-n-1-a-1-2-L-lim-n-a-n-2-2-2-2-L-

Question Number 4589 by FilupSmith last updated on 09/Feb/16

a_{n} = \sqrt{2 + a_{n - 1}}

a_{1} = \sqrt{2}

L = \lim_{n \to \infty} a_{n} = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}}

L = ?

Answered by Rasheed Soomro last updated on 09/Feb/16

a_{n} = \sqrt{2 + a_{n - 1}}

a_{1} = 2

a_{2} = \sqrt{2 + a_{1}} = \sqrt{2 + 2} = 2

a_{3} = \sqrt{2 + a_{2}} = \sqrt{2 + 2} = 2

T h i s s u g g e s t s t h a t

a_{n} = 2

H o w a_{n} = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}} ?

\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots .

A n s w e r f o r c o r r e c t e d q u e s t i o n

a_{n} = \sqrt{2 + a_{n - 1}}

a_{1} = \sqrt{2}

L = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}}

L^{2} = 2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}}

L^{2} = 2 + L

L^{2} - L - 2 = 0

(L - 2) (L + 1) = 0

L = 2 ∣ L = - 1

\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}} m a y n o t b e n e g a t i v e

a n d t h e r e i s a l s o c h a n c e o f e x t r a n e o u s

r o o t s, s o I t h i n k - 1 i s e x t r a n e o u s r o o t .

S o L = 2

Commented by FilupSmith last updated on 09/Feb/16

s o r r y i m e a n t

a_{1} = \sqrt{2}