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Show that the middle term in the expans...

Show that the middle term in the expansion of `(1+ x)^(2n)` is `(1.3.5…(2n-1) )/( n!). 2^(n) . x^(n)`, where `n in N`.

Text Solution

Verified by Experts

In the expansion of `(1+x)^(2n)` in total 2n+1 terms are present.
Thus, middle term will be, `​((2n+1)+1)/2=(2n+2​)/2=n+1​`
Thus, middle term will be `(n+1)th` term.
So,
`a​=(2n_C_n)​1^nx^(2n−n)`
`=(2n!​x^n)/(n!(2n−n)!)`
`=(2n(n!)1.3.5....(2n−1)​)/x^n`
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