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) = overset(n)underset(r=0)Sigma C_(r)x^(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_(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)+(C_(1))/(2)+......+(C_(n))/(n+1)=2

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)=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_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .

If (1+x)^n=sum_(r=0)^n C_r x^r , then prove that C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .

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) = overset(n)underset(r=0) Sigma C_(r) x^(r ) , show that C_(0)+(C_(1))/(2)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)