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1. $\sqrt{2 + \sqrt{2 + \sqrt{2 + . . . . . .}}}$ is equal to ?

A. $\sqrt{2}$

B. $2 \sqrt{2}$

C. 2

D. 3

Solution:

\begin{aligned} x = \sqrt{2 + \sqrt{2 + \sqrt{2 + . . . . . .}}} \\ \Rightarrow x^{2} = 2 + \sqrt{2 + \sqrt{2 + . . . . . . .}} \\ \Rightarrow x^{2} = 2 + x \\ \Rightarrow x^{2} - x - 2 = 0 \\ \Rightarrow x (x - 2) + 1 (x - 2) = 0 \\ \Rightarrow (x + 1) (x - 2) = 0 \\ \Rightarrow x = 2 \end{aligned}

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