`p*C_(0)+p^(2)(C_(1))/(2)+p^(3)(C_(2))/(3)+...+p^(n+1)*(C_(n))/(n+1)=((p+1)^(n+1)-1)/(n+1)`

C_(0)-(C_(1))/(2)+(C_(2))/(3)-......+(-1)^(n)(C_(n))/(n+1)=(1)/(n+1)

C_(0)-(C_(1))/(2)+(C_(2))/(3)-............(-1)^(n)(C_(n))/(n+1)=(1)/(n+1)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - (C_(1))/(2) + (C_(2))/(3) -…+ (-1)^(n) (C_(n))/(n+1) = (1)/(n+1) .

(C_(0))/(2)+(C_(1))/(3)+(C_(2))/(4)+(C_(3))/(5)+.......+(C_(n))/(n+2)=(1+n*2^(n+1))/((n+1)(n+2))

Prove that (i) C_(1)+2C_(2)+3C_(3)+……+nC_(n)=n.2^(n+1) (ii) C_(0)+(C_(1)/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)

(C_(1))/(C_(0))+2(C_(2))/(C_(1))+3(C_(3))/(C_(2))+.........+n(C_(n))/(C_(n-1))=(n(n+1))/(2)

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

Prove that (C_(0)+C_(1))(C_(1)+C_(2))(C_(2)+C_(3))(C_(n-1)+C_(n))=((n+1)^(n))/(n!)*c_(0)*C_(1)*C_(2).........

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