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NDA Math Definite Integrals

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For the next two (2) items that follow :

Given that

a_n = \int\limits_{0}^{\pi} \frac{\sin^2{(n+1)x}}{\sin 2x} dx

[NDA I 2016]

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Question : 37 of 62

Marks: +1, -0

Consider the following statements :

1. The sequence

\{a_{2 n}\}

is in AP with common difference zero.

2. The sequence

\{a_{2 n+1}\}

is in AP with common difference zero.

Which of the above statements is/are correct?

1 only
2 only
Both 1 and 2
Neither 1 nor 2

Solution:

a_{n} = \int\limits_{0}^{\pi} \frac{\sin^{2}((n+1)x)}{\sin(2x)} \, dx

Since it is a definite integral will have a definite value.The sequence

\{a_{2n}\}

is in AP with common difference.

Statement (1) is correct.

The sequence

a_{2n}+x

is also in AP with common difference.

Statement (2) is correct

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