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If 1 + x + x 2 25 = a 0 + a 1 x + a 2 x 2 + . … … . + a 50 x 50 , then a 0 + a 2 + a 4 + . … … . + a 50 is

Options

  1. Aeven
  2. Bodd and of the form 3 n
  3. Codd and of the form 3 n - 1
  4. Dodd and of the form 3 n + 1

Correct answer

A. even

Step-by-step solution

Putting x = 1 and x = - 1 and adding, we get, 2 a 0 + a 2 + a 4 + . … … . . + a 50 = 3 25 + 1 = 1 + 2 25 + 1 = C 25 0 + C 25 1 ⋅ 2 + C 25 2 ⋅ 2 2 + . … … + C 25 25 ⋅ 2 25 + 1 = 2 1 + C 25 1 + C 25 2 ⋅ 2 + . … … + C 25 25 ⋅ 2 24 ⇒ a 0 + a 2 + a 4 + . … … … + a 50 = 1 + 25 + C 2 25 ⋅ 2 + . … … + C 25 25 ⋅ 2 24 = 2 13 + C 25 2 + C 25 3 ⋅ 2 + . … … … + C 25 25 ⋅ 2 23 which is a

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