Nested radicals

Algebra Level 3

1 + 1 + 1 + ⋯ = x \large \sqrt{1+\sqrt{1+\sqrt{1+ \cdots}}} = x

Let the above equation be true for some real number x x . What is the value of x 2 + 1 x 2 \large x^2+\dfrac{1}{x^2} ?


The answer is 3.

Yash Jain by Yash Jain , 21, India

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1 solution

Yash Jain
Mar 21, 2017

1 + 1 + 1 + ⋯ = x \sqrt{1+\sqrt{1+\sqrt{1+ \cdots}}} = x

1 + x = x \sqrt{ 1+x } = x

Squaring both the sides,

1 + x = x 2 1+x = x^2

Dividing by x x as x ≠ 0 x \neq 0 ,

1 x + 1 = x \frac{1}{x}+1 = x

x − 1 x = 1 x-\frac{1}{x} = 1

Squaring both the sides,

x 2 + 1 x 2 − 2 = 1 x^2+\frac{1}{x^2}-2 = 1

x 2 + 1 x 2 = 3 x^2+\frac{1}{x^2} = \boxed{3}

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