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Show that the middle term in the expansi...

Show that the middle term in the expansion of `(1+x)^(2n)i s((1. 3. 5 (2n-1)))/(n !)2^n x^n ,w h e r en` is a positive integer.

Text Solution

Verified by Experts

As 2n is even, the middle term of the expansion `(1+x)^(2n)` is `((2n)/(2) + 1)` th or `(n+1)` th
`T_(n+1) = .^(2n)C_(n) x^(n) = ((2n)!)/(n!n!) x^(n)`
`= (1.2.3.4"….."(2n-2)(2n-1)(2n))/(n!n!)x^(n)`
`= ([1.3.5"…."(2n-1)][2.4.6"…."(2n)])/(n!n!) x^(n)`
` = ([1.3.5"...."(2n-1)]n!)/(n!n!)2^(n).x^(n)`
` = (1.3.5"....."(2n-1))/(n!) 2^(n)x^(n)`
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