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`

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

.^(n)C_(r)+2.^(n)C_(r-1)+.^(n)C_(r-2)=

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

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) - C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .

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

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

""^(n-2)C_(r)+2""^(n-2)C_(r-1)+""^(n-2)C_(r-2) equals :

If (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a+.^(n)C_(2)a^(2)+ . . +.^(n)C_(n)a^(n) , then prove that .^(n)C_(1)+2.^(n)C_(2)+3.^(n)3C_(3)+ . . .+n.^(n)C_(n)=n.2^(n-1) .

Prove that .(r+1)*^n C_r-r*^n C_r+ ... +(-1)^r.^n C_r=(-1)^r.^(n-2)C_rdot