Never Ending roots

Algebra Level 2

42 + 42 + 42 + ⋯ = ? \large \sqrt{42 + \sqrt{42 + \sqrt{ 42 +\sqrt{\cdots}}}} = \, ?

7 5 -6 8

Rajan Shah by Rajan Shah , 22, Nepal

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2 solutions

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Freddie Hand
Feb 5, 2017

Say x = 42 + 42 + 42 + . . . . . . x=\sqrt{42 + \sqrt{42 + \sqrt{ 42 + ......}}}

Then x 2 = 42 + 42 + 42 + 42 + . . . . . . x^{2}=42+\sqrt{42 + \sqrt{42 + \sqrt{ 42 + ......}}}

So x 2 − 42 = 42 + 42 + 42 + . . . . . . = x x^{2}-42=\sqrt{42 + \sqrt{42 + \sqrt{ 42 + ......}}}=x

Therefore, x 2 − x − 42 = 0 x^{2}-x-42=0

Solving, we get x = − 6 x=-6 or x = 7 x=7

Obviously, x = − 6 x=-6 is impossible, so x = 7 x=7 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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