Prove that : `1/2 nC_1-2/3nC_2+3/4nC_3-4/5nC_4+.....+((-1)^(n+1)n)/(n+1).nC_n=1/(n+1)`

If (1 + x)^n= ""^nC_0+""^nC_1x+""^nC_2x^2+.............+ ""^nC_n x^n , prove that : ""^nC_1-2""^nC_2+3""^nC_3-.......+(-1)^(n-1) n ""^nC_n=0 .

Prove that ""^nC_0-2*""^nC_1+3*""^nC_2-...+(-1)""^n(n+1)""^nC_n=0

Prove that ^nC_0-^nC_1+^nC_2-^nC_3+....+(-1)^r ^C_r+....=(-1)^(r-1) ^(n-1)C_(r-1)

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

Prove that (2nC_0)^2-(2nC_1)^2+(2nC_2)^2+.....+(2nC_2n)^2=(-1)^n2nC_n

Prove that (nC_(0))/(1)+(nC_(2))/(3)+(nC_(4))/(5)+(nC_(6))/(7)+....=(2^(n))/(n+1)

Prove that nC_1(nC_2)^2(n C_3)^3.......(n C_n)^n le ((2^n)/(n+1))^((n+1)C_2),AAn in Ndot

Prove that ""^nC_r+^nC_(r-1)=^(n+1)C_r

Prove that ""^nC_0+3*""^nC_1+5*""^nC_2+...+(2n+1)""^nC_n=(n+1)2^(n)

Prove that ""^nC_0+3""^nC_1+5""^nC_2+........+(2n+1)""^nC_n= (n+1)2^n .