If `(1+x)^n=c_0+C_1x+C_2x^2++C_n x^n `,using derivative prove that `C_1+2C_2++nC_n=n .2^(n-1)` `C_1-2C_2+3C_3++(-1)^(n-1)nC_n=0`

If (1+x)^n=c_0+C_1x+C_2x^2++C_n x^n , using derivtives prove t h a t C_1-2C_2+3C_3++(-1)^(n-1)nC_n=0

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0) C_(n) - C_(1) C_(n-1) + C_(2) C_(n-2) - …+ (-1)^(n) C_(n) C_(0) = 0 or (-1)^(n//2) (n!)/((n//2)!(n//2)!) , according as n is odd or even .

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

If (1+x)^n=sum_(r=0)^n C_r x^r , then prove that C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .

Find the value of C_0 - [C_1 -2.C_2+ 3.C_3-……..+(-1)^(n-1).n.C_n]

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

If (1+x)^n = C_0 + C_1x + C_2x^2 + ……..+C_n.x^n then find C_1 - C_3 + C_5 + ……

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

If (1+x)^n = C_0 + C_1x + C_2x^2 + ……..+C_n.x^n then find C_0 - C_2 + C_4 - C_6 + …….