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If the sum of the series 1 + 3 2 + 5 4 + 7 8 + . . . . + 2 n - 1 2 n - 1 is f n , then the value of f 8 is

Options

  1. A4 + 12 2 5
  2. B5 + 13 2 7
  3. C6 - 19 2 7
  4. D5 - 13 2 7

Correct answer

C. 6 - 19 2 7

Step-by-step solution

S n = 1 + 3 2 + 5 4 + 7 8 + . . . . . + 2 n - 1 2 n - 1 1 2 S n = 1 2 + 3 4 + 5 8 + . . . . . + 2 n - 3 2 n - 1 + 2 n - 1 2 n Subtracting, we get, 1 2 S n = 1 + 2 2 + 2 4 + 2 8 + . . . . . + 2 2 n - 1 - 2 n - 1 2 n 1 2 S n = 1 + 2 1 2 + 1 4 + 1 8 + . . . . . n - 1 t e r m s - 2 n - 1 2 n 1 2 S n = 1 + 2 1 2 1 - 1 2 n - 1 1 - 1 2 - 2 n - 1 2 n 1 2 S n = 1 + 2 1 - 1 2 n - 1 - 2 n - 1 2 n = 1 + 2 - 1 2 n - 2 - 2 n - 1 2 n f n = S n = 6 - 1 2 n - 3 - 2 n - 1 2 n - 1 ⇒ f 8 = 6 - 1 2 5 - 15 2 7 f 8 = 6 - 4 + 15 2 7

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