`(C_0+C_1)(C_1+C_2)(C_2+C_3)(C_3+C_4)...........(C_(n-1)+C_n)=(C_0C_1C_2.....C_(n-1)(n+1)^n)/(n!)`

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

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 that (C_0+C_1)(C_1+C_2)(C_2+C_3)...(C_(n-1)+C_n)=(C_0C_1C_2...C_(n-1)(n+1)^n)/(n!)

( (C_0 +C_1)(C_1 + C_2)(C_2 +C_3).....(C_(n-1) +C_n))/(C_0C_1C_2....C_n)= (A) (n+1) ^n/n (B) (n+1)^(n-1)/n! (C) (n+1)^n/(n!)(D) (n+1)/(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!) .

((C_0 + C_1)(C_1 + C_2)(C_2 + C_3)………(C_(n-1) + C_n) )/(C_0C_1C_2…C_n)

C_1/C_0+2C_2/C_1+3C_3/C_2+............+nC_n/C_(n-1)=(n(n+1))/2

C_1/C_0+2C_2/C_1+3C_3/C_2+............+nC_n/C_(n-1)=(n(n+1))/2