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If 1 + x + 2 x 2 20 = a 0 + a 1 x + a 2 x 2 + . . . . + a 40 x 40 , then the value of a 0 + a 2 + a 4 + . . . . . + a 38 is

Options

  1. A2 20 2 20 - 1
  2. B2 20 2 20 + 1
  3. C2 39 - 2 19
  4. D2 39 + 2 19

Correct answer

C. 2 39 - 2 19

Step-by-step solution

In the given identity, substituting x = 1 and x = - 1 on both the sides and adding, we get, 2 a 0 + a 2 + a 4 + . . . . . + a 40 = 4 20 + 2 20 = 2 40 + 2 20 ⇒ a 0 + a 2 + a 4 + . . . . + a 38 + a 40 = 2 39 + 2 19 But, a 40 is the coefficient of x 40 which is 2 20 . Therefore, a 0 + a 2 + a 4 + . . . . . + a 38 + 2 20 = 2 39 + 2 19 ⇒ a 0 + a 2 + a 4 + . . . . . + a 38 = 2 39 + 2 19 - 2 . 2 19 = 2 39 - 2 19

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