[" Given that "C_(1)+2C_(2)x+3C_(3)x^(2)+cdots+2nC_(2n)],[=2n(1+x)^(2n-1)," where "C_(r)=(2n)!/[r!(2n-r)!];r=0],[....2n," then prove that "C_(1)^(2)-2C_(2)^(2)+3C_(3)^(2)-...-2],[=(-1)^(n)nC_(n)]

Given that C_(1)+2C_(2)x+3C_(3)x^(2)+...+2nC_(2n)x^(2n-1)=2n(1+x)^(2n-1),whereC_(r)=(2n)!/[r!(2n-r)!];r=0,1,2 then prove that C_(1)^(2)-2C_(2)^(2)+3C_(3)^(2)-...-2nC_(2n)^(2)=(-1)^(n)nC_(n).

Give that : C_1+2C_2 x+3C_3 x^2+.....+2n. C_(2n). x^(2n-1)=2n(1+x)^(2n-1) , where C_r=((2n)!)/(r !(2n-r)!) , r=0,1,2, ,2n , then prove that C_1^2-2C_2^2+3C_3^2-..........-2n C_(2n)^2=(-1)^n . nC_n .

Given that C_1+2C_2x+3C_3x^2++2nC_(2n)x^(2n-1)=2n(1+x)^(2n-1), w h e r eC_r=(2n)!//[r !(2n-r)!]; r=0,1,2, ,2n , then prove that C1 2-2C2 2+3C3 2--2n C2n2=(-1)^nn C_ndot

Prove that: C_(1) + 2C_(2) + 3C_(3) + 4C_(4) +….. + nC_(n) = n.2^(n-1)

Prove that C_(0)2^(2)C_(1)+3C_(2)4^(2)C_(3)+...+(-1)^(n)(n+1)^(2)C_(n)=0 where C_(r)=nC_(r)

c_(1)^(2)+2C_(2)^(2)+3C_(3)^(2)+....+nC_(n)^(2)=((2n-1)!)/ ([(n-1)!^(2)))

Prove that sum_(r=0)^(2n) r.(""^(2n)C_(r))^(2)= 2.""^(4n-1)C_(2n-1) .

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

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