Prove that : Prove that
`(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 : Prove that C_(0)+3.C_(1)+3_(2).C_(2)+……+3^(n).C_(n)=4^(n)

With usual notations prove that C_1/C_0 + 2. C_2/C_1 + 3.C_3/C_2 + ……+n.(C_n)/(C_(n-1)) = (n(n +1))/(2) Hence prove that (15C_1)/(15C_0) + 2.(15C_2)/(15C_1) + 3. (15C_3)/(15C_2) +……..+ 15. (15C_15)/(15C_14) = 120

Show that C_(0) +(C_(0) +C_(1))+(C_(0)+C_(1)+C_(2)) +…+(C_(0) +C_(1)+…+C_(n)) =(n+2).2^(n-1)

Prove that : If n is a positive integer and x is any nonzero real number, then prove that C_(0)+C_(1)(x)/(2)+C_(2).(x^(2))/(3)+C_(3).(x^(3))/(4)+….+C_(n).(x^(n))/(n+1)=((1+x)^(n+1)-1)/((n+1)x)

Prove that following (C_(1))/(2)+(C_(3))/(4)+(C_(5))/(6)+(C_(7))/(8)+……=(2^(n)-1)/(n+1)

Prove that : Prove that 2.C_(0)+7.C_(1)+12.C_(2)+……..+(5n+2)C_(n)=(5n+4)2^(n-1).

Prove that C_(0)-(1)/(3)*C_(1)+(1)/(5)*C_(2) - …+(-1)^(n)*(1)/(2n+1)C_(n) =(2^(2n)(n !)^(2))/((2n+1)!)

Prove that (C_0)/(1)+ (C_2)/(3) + (C_4)/(5) + (C_6)/(7) +…….= (2^n)/(n+ 1)