If `(1+x)^n=underset(r=0)overset(n)C_(r)x^r` then prove that `C_(1)^2+2.C_(2)^(2)+3.C_(3)^2 +…….+n.C_(n)^(2)=((2n-1)!/((n-1)!)^2`

Explanation

`(1+x)^(n)=C_0+C_(1)x+C_(2)x^2+C_(2)x^3+......+C_(n)x^n..........(i)`
Differentiating both the sides ,w.r.t x, we get
`n(1+x)^(n-1)=C_(1)+2C_(2)x+3C_(2)x^2+....+n.C_(n)x^(n+1)....(ii)`
also ,we have
`(x+1)^(n)=C_(0)x^n+C_(1)x^(n-1)+C_(2)x^(n-2)+.....+C_(n).....(iii)`
Equating the coefficients of `x^(n-1)` we get
`(C_(1)^(2)+2C_(2)^(2)+3C_(3)^2+.....+C_(n)x^(n-1))(C_(0)x^(n-1)+C_(2)x^(n-2)+......+C_(n))=n(1+x)^(2n-1)`
Equating the coffiecients of `x^(n-1)` we get
`C_(1)^2+2C_2^2+3C_3^2+.....+n.C_n^2=n.""^(2n-1)C_(n-1)=((2n-1)!)/(((n-1)!)^2)`