Question

$\sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2} } } }$
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Answer 1

$\text{Let us assume that,}\\\\y=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}}\\\\\\\implies y=\sqrt{2+y}\\\\\implies y^2=2+y\\\\y^2-y-2=0\\\\y^2-2y+y-2=0\\\\y(y-2)+(y-2)=0\\\\(y+1)(y-2)=0\\\\\implies y=-1\:or\:y=2\\\\\text{Sum of square roots cant be a negative number}\implies y=2.\\\\\\y=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}}=2$