If `(1+x)^n=sum_(r=0)^n C_rx^r` then prove that `C_1+2C_2+3C_3+.....+nC_n=n2^(n-1)`

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r), then prove that C_(1)+2c_(2)+3C_(1)+...+nC_(n)=n2^(n-1)...

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r) then prove that C_(0)+2C_(1)+3C_(2)+.....+(n+1)C_(n)=2^(n-1)(n+2)

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r) then prove that C_(0)+(C_(1))/(2)+......+(C_(n))/(n+1)=2

If (1+x)^n=sum_(r=0)^n C_rx^r then prove that sum_(r=0)^n (C_r)/((r+1)2^(r+1))=(3^(n+1)-2^(n+1))/((n+1)2^(n+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=C_0+C_1x+C_2x^2+_____+C_nx^n , prove that C_1+2C_2+3C_3+_____+ ^nC_n=n2^(n-1)

If (1+x)^n=C_0+C_1x+C_2x^2+ _____+C_nx^n, Prove that, C_1+2C_2+3C_3+_____+nC_n=n.2^(n-1) .

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r), show that (C_(0))/(2)+(C_(1))/(3)+(C_(2))/(4)+...+(C_(n))/(n+2)=(n*2^(n+1)+1)/((n+1)(n+2))

If (1+x)^(n)=C_(0)+C_(1).x+C_(2).x^(2)+C_(3).x^(3)+......+C_(n).x^(n), then prove that C_(0)+2C_(1)+4C_(2)+6C_(3)+...+2n.C_(n)=1+n*2^(n)