UPSC CDS 1 2023 Math Paper

Concept:This is an infinite nested radical, which can be expressed as an algebraic equation: $x = \sqrt{4x}$.Explanation:Given $x = \sqrt{4\sqrt{4\sqrt{4\sqrt{4\ldots}}}}$, the pattern repeats indefinitely. So the entire expression under the first square root is $4x$, giving $x = \sqrt{4x}$. Square both sides: $x^2 = 4x$. Bring all terms to one side: $x^2 - 4x = 0$. Factor: $x(x - 4) = 0$. Thus $x = 0$ or $x = 4$. Since $x$ is a square root of a positive value, $x > 0$, so $x = 0$ is discarded. Therefore, the required value is $x = 4$.Answer:$4$