Show that `4C_0 +4C_1 +4C_2 +.....+4C_n>(2^(4n))/(n^3),w h e r e^n C_r=n !//[r !(n-r)!]dot`

Given that C_1+2C_2x+3C_3x^2++2nC_(2n)x^(2n-1)=2n(1+x)^(2n-1), w h e r eC_r=(2n)!//[r !(2n-r)!]; r=0,1,2, ,2n , then prove that C1 2-2C2 2+3C3 2--2n C2n2=(-1)^nn C_ndot

Prove that C_0-2^2C_1+3^2C_2-4^2C_3+...+(-1)^n(n+1)^2xxC_n=0w h e r eC_r=^n C_r .

If n >2, then prove that C_1(a-1)-C_2xx(a-2)++(-1)^(n-1)C_n(a-n)=a ,w h e r eC_r=^n C_rdot

Find the sum 1C_0+2C_1+3C_2++(n+1)C_n ,w h e r eC_r=^n C_rdot

Find the sum 1xx2xxC_1+2xx3C_2+ +n(n+1)C_n ,w h e r eC_r=^n C_rdot

Find the sum C_0-C_2+C_4-C_6+ ,w h e r eC_r=^6C_r=^n C_r .

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

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

Prove that (r+1)^n C_r-r^n C_r+(r-1)^n C_2-^n C_3++(-1)^r^n C_r = (-1)^r^(n-2)C_rdot

If C_(r) = .^(n)C_(r) then prove that (C_(0) + C_(1)) (C_(1) + C_(2)) "….." (C_(n-1) + C_(n)) = (C_(1)C_(2)"…."C_(n-1)C_(n))(n+1)^(n)//n!