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Irrational Numbers on the Number Line

Find the valu...

Question

Find the value of $\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{. . . to \infty}}}}}$

A

1

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B

2

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C

3

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D

4

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Solution

The correct option is B $2$
Take $x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{. . . t o \infty}}}}}$
Squaring both sides, we get
$x^{2} = 2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{. . . . t o \infty}}}} = 2 + x$
or $x^{2} - x - 2 = 0$ or $(x + 1) (x - 2) = 0$
$∴ x = - 1$ or 2
However, x cannot be negative since the given expression is positive. Therefore, $x = - 1$ is ruled out.Therefore, $x = 2$ .

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