Find the sum `1xx2xxC_1+2xx3C_2+` `+n(n+1)C_n ,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

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

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 .

Find the sum 3^n C_0-8^n C_1+13^n C_2 - 18^n C_3+..

Find the sum sum_(r=1)^n r^2(^n C_r)/(^n C_(r-1)) .

Find the sum of sum_(r=1)^n(r^n C_r)/(n C_(r-1) .

Find the sum sum_(r=0)^n^(n+r)C_r .

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