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+2C(2)*a+3*C(3)*a^(2)+...+2n*C(2n)*a^(2n...

+2C_(2)*a+3*C_(3)*a^(2)+...+2n*C_(2n)*a^(2n-1)=

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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).

If (1+x)^(n)=C_(0)+C_(1)+x+C_(2)x^(2)+...+C_(n) x^(n) Show that C_(1)^(2)+2*C_(2)^(2)+3*C_(3)^(2)....+n*C_(n)^(2)=((2n-1)!)/([(n-1)!]^(2))

Prove that C_(1)^(2)-2*C_(2)^(2)+3*C_(3)^(2)-…-2n*C_(2n)^(2)=(-1)^(n)n*C_(n)

Prove that C_(1)^(2)-2*C_(2)^(2)+3*C_(3)^(2)-…-2n*C_(2n)^(2)=(-1)^(n)n*C_(n)

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

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 .

With usual notations prove that C_1+2.C_2x + 3.C_3x^2 + ……+2n.C_(2n) x^(2n - 1) = 2n (1 + x)^(2n -1)

With usual notations prove that C_1+2.C_2x + 3.C_3x^2 + ……+2n.C_(2n) x^(2n - 1) = 2n (1 + x)^(2n -1)

If (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a+.^(n)C_(2)a^(2)+ . . +.^(n)C_(n)a^(n) , then prove that .^(n)C_(1)+2.^(n)C_(2)+3.^(n)3C_(3)+ . . .+n.^(n)C_(n)=n.2^(n-1) .