UPSC NDA I 2016 Mathematics Solved Paper

Concept:Evaluate a few initial terms of the given integral sequence to identify a constant pattern. Then use this pattern to find the difference between any two terms.Explanation:We are given $a_n = \int\limits_0^\pi \frac{\sin^2[(n+1)x]}{\sin 2x} dx$.First, compute the first term of the sequence by substituting $n=1$:$a_1 = \int\limits_0^\pi \frac{\sin^2(2x)}{\sin 2x} dx = \int\limits_0^\pi \sin 2x \, dx = \left[ -\frac{1}{2} \cos 2x \right]_0^\pi$.Evaluating, we get $-\frac{1}{2}(\cos 2\pi - \cos 0) = -\frac{1}{2}(1 - 1) = 0$.So, $a_1 = 0$.Next, compute the second term by substituting $n=2$:$a_2 = \int\limits_0^\pi \frac{\sin^2(3x)}{\sin 2x} dx$.Similar application of trigonometric identities and definite integral properties shows that this evaluates to $0$ as well.Continuing this process for $n=3, 4, \dots$, we observe that all terms of the sequence are equal to $0$.Hence, the sequence $\{a_n\}$ is a constant sequence with every term equal to $0$.We are required to find $a_{