How long is this 6?

Algebra Level 2

6 + 6 + 6 + . . . = ? \large \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{ ... }}}} = \ ?

Can't be solved 2 6 1 3

Syed Hamza Khalid by Syed Hamza Khalid , 17, France

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1 solution

Syed Hamza Khalid
Oct 15, 2017

Supose that:

x = 6 + 6 + 6 + 6 + . . . \large{x = \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{ ... }}}}}}

Then:

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

→ x 2 − x − 6 = 0 \to x^2 - x - 6 = 0

Solving it, we get: x = 3 or x = -2 \large \color{#3D99F6} \text{x = 3 or x = -2}

But since we don't have − 2 -2 as the choices, therefore the answer is:

x = 3 \huge \color{#D61F06} \boxed{x \text{= 3}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

I guess you should mention x = − 2 x=-2 is an extraneous root.

Aman thegreat - 3 years, 7 months ago

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Ok! Then I will have this solution edited. Thanks for noting this down and I would like you to upvote my solution. Hope you enjoy!

Syed Hamza Khalid - 3 years, 7 months ago

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