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Manipal MET2014MathematicsBinomial Theorem

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

Options

  1. An! (n!)^2
  2. B(2 n!) (n!)^2
  3. C(2 n-1)! n!
  4. D(2 n)! n!

Correct answer

B. (2 n!) (n!)^2

Step-by-step solution

The given expression is aligned (x^2+ 1 x^2 +2 )^n & = [ (x+ 1 x )^2 ]^n & = (x+ 1 x )^ 2 n aligned The binomial contains (2 n+1) terms in its expansion. The middle term is T_ n+1 . aligned T_ n+1 & = ^ 2 n C_n(x)^ 2 n-n ( 1 x )^n & = ^ 2 n C_n & = (2 n)! (2 n-n)!n! = (2 n)! (n!)^2 aligned

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