Find-the-value-of-6-6-6-6-6-

Question Number 4825 by sanusihammed last updated on 16/Mar/16

F i n d t h e v a l u e o f

\sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6}}}}}

Commented by prakash jain last updated on 16/Mar/16

Assuming the series goes infinitely

x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \dots}}}}

x = \sqrt{6 + x} \Rightarrow x = 3

Now if the value is only for 6 nested roots .

\sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6}}}}} = 2.997 \approx 3

Commented by 123456 last updated on 16/Mar/16

y = - \sqrt{6 - y}

y^{2} = 6 - y

y^{2} + y - 6 = 0

Δ = {(1)}^{2} - 4 (1) (- 6) = 1 + 24 = 25

y = \frac{- 1 \pm 5}{2}

y_{1} = \frac{- 1 - 5}{2} = - \frac{6}{2} = - 3

y_{2} = \frac{- 1 + 5}{2} = \frac{4}{2} = 2

Answered by Rasheed Soomro last updated on 16/Mar/16

L e t x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6}}}}} \dots .

x^{2} = 6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6}}}}} \dots . .

x^{2} = 6 + x \Rightarrow x^{2} - x - 6 = 0

x = \frac{- (- 1) \pm \sqrt{{(- 1)}^{2} - 4 (1) (- 6)}}{2 (1)} = \frac{1 \pm \sqrt{25}}{2}

x = 3, - 2

- 2 i s e x t r a n e o u s r o o t .

∴ x = 3