prove-that-n-x-n-2-x-n-2-x-n-n-2-x-2n-n-2-Ramanujan-s-nested-radikal-

Question Number 154034 by amin96 last updated on 13/Sep/21

p r o v e t h a t

n + x = \sqrt{n^{2} + x \sqrt{n^{2} + (x + n) \sqrt{n^{2} + (x + 2 n) \sqrt{n^{2} \dots}}}}

{R a m a n u j a n}^{'} s n e s t e d r a d i k a l

Answered by mr W last updated on 13/Sep/21

{(n + x)}^{2} = n^{2} + 2 x n + x^{2}

{(n + x)}^{2} = n^{2} + x (n + x + n) \dots (i)

\Rightarrow n + x = \sqrt{n^{2} + x (n + x + n)} \dots (i i)

i n (i) r e p l a c e x w i t h x + n,

{(n + x + n)}^{2} = n^{2} + (x + n) (n + x + 2 n)

\Rightarrow (n + x + n) = \sqrt{n^{2} + (x + n) (n + x + 2 n)}

p u t t h i s i n t o (i i),

\Rightarrow n + x = \sqrt{n^{2} + x \sqrt{n^{2} + (x + n) (n + x + 2 n)}} \dots (i i i)

i n (i) r e p l a c e x w i t h x + 2 n,

{(n + x + 2 n)}^{2} = n^{2} + (x + 2 n) (n + x + 3 n)

\Rightarrow (n + x + 2 n) = \sqrt{n^{2} + (x + 2 n) (n + x + 3 n)}

p u t t h i s i n t o (i i i),

\Rightarrow n + x = \sqrt{n^{2} + x \sqrt{n^{2} + (x + n) \sqrt{n^{2} + (x + 2 n) (n + x + 3 n)}}} \dots (i v)

a n d s o o n s o o n s o o n \dots .

Commented by amin96 last updated on 13/Sep/21

g o o d w o r k s i r

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