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`.

Answer

N/a

Explanation

No of terms in the expansion of `(x+a)^(2n) =2n+1`(odd) ltbr. `:. " Middle term " =((2n+1)+1)/(2)`th term
`T_(n+1)=^(2n)C_(n).x^((2n-n).a^(n)=(|ul2n)/(|uln|uln).x^(n).a^(n)`
`=((1.2.3...(2n-1)][2.4.6...(2n)])/(|uln|uln).x^(n).a^(n)`
`[[1.3.5...(2n-1)][2^(n).[1.2.3....n)]]/(|uln|uln).x^(n).a^(n)`
`=[[1.3.5....(2n-1)][.2^(n).[1.2.3....n]]]/(|uln.[1.2.3....n]].x^(n).a^(n)`
`=[[1.3.5...(2n-1)]]/(|uln)2^(n).x^(n).a^(n)` Hence Proved