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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Definite Integration

NDA 2016 · Mathematics · Definite Integration

Given that a_ n =∫_0^π ^2\ (n+1) x\ 2 x dx Consider the following...

Given that an=∫0πsin⁡2{(n+1)x}sin⁡2xdx\mathrm{a}_{\mathrm{n}}=\int_0^\pi \frac{\sin ^2\{(\mathrm{n}+1) \mathrm{x}\}}{\sin 2 \mathrm{x}} \mathrm{dx} Consider the following statements: 1. The sequence {a2n}\left\{a_{2 n}\right\} is in AP with common difference zero. 2. The sequence {a2n+1}\left\{\mathrm{a}_{2 \mathrm{n}+1}\right\} is in AP with common difference zero. Which of the above statements is/are correct?

Options

  1. A1 only
  2. B2 only
  3. CBoth 1 and 2
  4. DNeither 1 nor 2

Correct answer

C. Both 1 and 2

Step-by-step solution

an=∫0πsin⁡2{(n+1)x}sin⁡2xdxa_n=\int_0^\pi \frac{\sin ^2\{(n+1) x\}}{\sin 2 x} d x Since it is a definite integral will have a definite value. The sequence {a2n}\left\{a_{2 n}\right\} is in AP with common difference. Statement (1) is correct. The sequence {a2n+1}\left\{a_{2 n+1}\right\} is also in AP with common difference. Statement (2) is correct.
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