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TIFR2010-Maths-A-7

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Let $$x = \sqrt{7 + \sqrt{7 + \sqrt{7 \ldots}}}$$

Then, we can write $x$ recursively as:

$$x = \sqrt{7+x}$$

Solving this equation, we get:

$$\begin{align}x^2 &= 7+x\\[2em]x&=\frac{-(-1)\pm\sqrt{(-1)^2-4\cdot(-7)}}{2}\\[2em]x &= \frac{1\pm\sqrt{29}}{2}\end{align}$$

Hence, option D is the correct answer.

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