First slide

Special Series in Sequences and Series

Question

Let $x = \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \dots upto \infty}}}}$ and $y = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots upto \infty}}}},$ then $x - y$ equals

Moderate

A

0
B

1
C

2
D

none of these

Solution

We have $x^{2} = 2 x$ and $y^{2} = 2 + y$
$\Rightarrow x (x - 2) = 0$ and $(y - 2) (y + 1) = 0$
$\begin{array}{l} \Rightarrow x = 2, y = 2 [∵ x, y > 0] \\ \Rightarrow x - y = 0 \end{array}$

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