`(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))`

(C_(0))/(1.2)+(C_(1))/(2.3)+(C_(2))/(3.4)+......*(C_(n))/((n+1)(n+2))=

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))

(C_(0))^(2)+2(C_(1))^(2)+3(C_(2))^(2)+4(C_(3))^(2)...+(n+1)(c_(n))^(2)

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

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

If C_(0),C_(1),C_(2),...,C_(n) denote the binomial coefficientsin the expansion of (1+x)^(n), then (C_(0))/(2)-(C_(1))/(3)+(C_(2))/(4)-(C_(3))/(5)+......+(-1)^(n)(C_(n))/(n+2)=

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

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

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