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COMEDK2022MathematicsBinomial Theorem

In the expansion of (1-3 x+3 x^2-x^3 )^ 2 n , the middle term is

Options

  1. A(n+1) th term
  2. B(2 n+1) th term
  3. C(3 n+1) th term
  4. DNone of these

Correct answer

C. (3 n+1) th term

Step-by-step solution

The given expression is (1-3x+3x^2-x^3 )^ 2n . Observe that 1-3x+3x^2-x^3 = (1-x)^3 . Substituting this into the expression, we get ((1-x)^3 )^ 2n = (1-x)^ 6n . The expansion of (1-x)^ 6n has 6n+1 terms. The middle term of an expansion with N terms is the ( N+1 2 ) th term. Here, N = 6n+1 , so the middle term is the ( 6n+1+1 2 ) th term = ( 6n+2 2 ) th term = (3n+1) th term. Answer: (3 n+1) th term

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