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Let a n = ∫ 0 π 2 1 - cos ⁡ 2 n x 1 - cos ⁡ 2 x d x , then a 1 , a 2 , a 3 , . . . . . . are in

Options

  1. AArithmetic Progression
  2. BGeometric Progression
  3. CHarmonic Progression
  4. DArithmetic Geometric Progression

Correct answer

A. Arithmetic Progression

Step-by-step solution

a n + 2 + a n - 2 a n + 1 = ∫ 0 π 2 1 - cos ⁡ 2 n + 2 x 1 - cos ⁡ 2 x d x + ∫ 0 π 2 1 - cos ⁡ 2 n x 1 - cos ⁡ 2 x d x - 2 ∫ 0 π 2 1 - cos ⁡ 2 n + 1 x 1 - cos ⁡ 2 x d x = ∫ 0 π / 2 1 - cos ⁡ 2 n + 4 x + 1 - cos ⁡ 2 n x - 2 + 2 cos ⁡ 2 n + 2 x 1 - cos ⁡ 2 x d x = ∫ 0 π 2 2 cos ⁡ 2 n + 2 x - cos ⁡ 2 n + 4 x + cos ⁡ 2 n x 1 - cos ⁡ 2 x d x = ∫ 0 π 2 2 cos ⁡ 2 n + 2

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