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AP EAMCET2016MathematicsBinomial Theorem

If the middle term in the expansion of (1+x)^ 2 n is the greatest term, then x lies in the interval

Options

  1. A( n n+1 , n+1 n )
  2. B( n+1 n , n n+1 )
  3. C(n-2, n)
  4. D(n-1, n)

Correct answer

A. ( n n+1 , n+1 n )

Step-by-step solution

In the expansion of (1+x)^ 2 n , middle term is ^ 2 n C _n x^n . Since, middle term is the greatest term. ^ 2 n C _n x^n> ^ 2 n C _ n-1 x^ n-1 and ^ 2 n C _r x^n> ^ 2 n C _ n+1 x^ n+1 aligned & x> ^ 2 n C _ n-1 ^ 2 n C _n and x < ^ 2 n C _n ^ 2 n C _ n+1 & Hence, x ( ^ 2 n C _n ^ 2 n C _ n+1 , ^ 2 n C _n ^ 2 n C _ n+1 ) aligned Hence, x ( ^ 2 n C _n ^ 2 n C _ n+1 , ^ 2 n C _n ^ 2 n C _ n+1 ) aligned & x ( array c (2 n) ! (n-1) !(2 n-n+1) n !(2 n-n) ! (2 n) ! (2 n) ! n !(2 n-n) ! (n-1) !(2 n-n+1) (2 n) ! array ) & x

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