[" (6) "C_(1)+C_(2)+C_(3)+......" in "],[" (7) "C_(0)+2C_(1)+3C_(2)+4C_(3)+...+(n+1)C_(n)=(n+2)2^(n-1)]

2C_(0)+5C_(1)+8C_(2)++(3n+2)C_(n)=(3n+4)2^(n-1)

C_(1) + 4.C_(2) + 7.C_(3) +......+(3n - 2).C_(n) =

C_(1)+4.C_(2)+=7.C_(3)+......+(3n-2).C_(n)=

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

If C_(0),C_(1),C_(2)…….,C_(n) are the combinatorial coefficient in the expansion of (1+x)^n, n, ne N , then prove that following C_(1)+2C_(2)+3C_(3)+..+n.C_(n)=n.2^(n-1) C_(0)+2C_(1)+3C_(2)+......+(n+1)C_(n)=(n+2)C_(n)=(n+2)2^(n-1) C_(0),+3C_(1)+5C_(2)+.....+(2n+1)C_n =(n+1)2^n (C_0+C_1)(C_1+C_2)(C_2+C_3)......(C_(n-1)+C_n)=(C_0.C_1.C_2....C_(n-1)(n+1)^n)/(n!) 1.C_0^2+3.C_1^2+....+ (2n+1)C_n^2=((n+1)(2n)!)/(n! n!)

Prove the following: C_(0)+2*C_(1)_+3*C_(2)+…(n+1)*C_(n)=(n+2).2^(n-1)

(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - 2C_(1) + 3C_(2) - 4C_(3) + … + (-1)^(n) (n+1) C_(n) = 0

(1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - 2C_(1) + 3C_(2) - 4C_(3) + … + (-1)^(n) (n+1) C_(n) = 0