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COMEDK2012MathematicsBinomial Theorem

The middle term in the expansion of (1+x)^ 2 n is

Options

  1. A1 3 5 (2 n-1) n x^ n
  2. B1 3 5 (2 n-1) n ! 2^ n-1 x^ n
  3. C1 3 5 (2 n) n ! x^ n
  4. D1 3 5 (2 n-1) n ! 2^ n x^ n

Correct answer

D. 1 3 5 (2 n-1) n ! 2^ n x^ n

Step-by-step solution

The middle term in the expansion of (1+x)^ 2 n is ( 2 n 2 +1 ) th term i.e. (n+1) th term. aligned &= (2 n) ! n ! n ! x^ n &= 2 n(2 n-1)(2 n-2) 3 2 1 n(n-1)(n-2) 3 2 1 ² x^ n &= (2 n(2 n-2)(2 n-4) 4.2 (2 n-1)(2 n-3) (3 1) n(n-1)(n-2) 3 2 1 ² x^ n &= 2^ n n(n-1)(n-2) 3 2 1 ((2 n-1)(2 n-3) 3 1 n(n-1)(n-2) 3 2 1 ² x^ n &= 2^ n (2 n-1)(2 n-3) 3 1 n(n-1)(n-2) 3 2 1 x^ n &= (2 n-1)(2 n-3) 3 1 n ! x^ n 2^ n aligned

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