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JEE Advanced2022MathematicsDefinite IntegrationActual

For positive integer n , define f n = n + 16 + 5 n - 3 n 2 4 n + 3 n 2 + 32 + n - 3 n 2 8 n + 3 n 2 + 48 - 3 n - 3 n 2 12 n + 3 n 2 + … + 25 n - 7 n 2 7 n 2 . Then, the value of lim n → ∞ f n is equal to

Options

  1. A3 + 4 3 log e 7
  2. B4 - 3 4 log e 7 3
  3. C4 - 4 3 log e 7 3
  4. D3 + 3 4 log e 7

Correct answer

B. 4 - 3 4 log e 7 3

Step-by-step solution

Given f n = n + 16 + 5 n - 3 n 2 4 n + 3 n 2 + 32 + n - 3 n 2 8 n + 3 n 2 + … . + 25 n - 7 n 2 7 n 2 = 16 + 5 n - 3 n 2 4 n + 3 n 2 + 1 + 32 + n - 3 n 2 8 n + 3 n 2 + 1 + … … + 25 n - 7 n 2 7 n 2 + 1 f n = 9 n + 16 4 n + 3 n 2 + 9 n + 32 8 n + 3 n 2 + … … + 25 n 7 n 2 i.e. f n = ∑ r = 1 n 9 n + 16 r 4 r n + 3 n 2 = 1 n ∑ r = 1 n 9 + 16 r n 4 r n + 3 Now, lim n → ∞ f n = ∫ 0 1 9 + 16 x 4 x + 3 d x = ∫ 0 1 16 x + 12 - 3 4 x + 3 d x = 4 x − 3 4 ln

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