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KVPY2010MathematicsBinomial Theorem

Arrange the expion of ( x ^ 1 / 2 + 1 2 x ^ 1 / 4 )^ n in decreasing powers of x . Suppose the coefficient of the first three terms form an arithmetic progression. Then the number of terms in the expion having integer powers of x is-

Options

  1. A1
  2. B2
  3. C3
  4. Dmore than 3

Correct answer

C. 3

Step-by-step solution

T_ r+1 = n n (x^ 1 / 2 )^ n-r 1 (2 x^ 1 / 4 )^ r = n r 2^ r x^ 2 n-3 r 4 array l T ₁, ~T ₂, ~T ₃ AP 2 ^ n C ₁ 2 = ^ n C ₀+ ^ n C ₂ 2² n -1= n ( n -1) 8 n =8 16-3 r 4 = Integers c =0,4,8 array

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