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Manipal MET2012MathematicsBinomial Theorem

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

Options

  1. An [(n / 2)!]^2
  2. B- (2 n!) [(n / 2)!]^2
  3. C1 3 5 (2 n+1) n! 2^n
  4. D(2 n)! (n!)^2

Correct answer

D. (2 n)! (n!)^2

Step-by-step solution

(x^2+ 1 x^2 +2 )^n= [ (x+ 1 x )^2 ]^n= (x+ 1 x )^ 2 n The number of terms in the expansion of (x+ 1 x )^ 2 n is 2 n+1 which is odd. Therefore, (n+1) th term will be middle term. aligned T_ n+1 & = ^ 2 n C_n(x)^ 2 n-n ( 1 x )^n & = ^ 2 n C_n x^n 1 x^n & = ^ 2 n C_n= (2 n)! n!n! = (2 n)! (n!)^2 aligned

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