If `(1+x)^n = C_0 + C_1x + C_2x^2 +….+C_nx^n`, then prove that `C_1+2C_2 + 3C_3+..+nCn=n 2^(n-1)`

If (1+x)^n = C_0 +C_1x+C_2x^2+……… +C_nx^n , then prove that : C_1 -2C_2+………….+(-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_(1) + 2C_(2) + 3C_(3) + ...+ n""C_(n) = n*2^(n-1)

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove that : C_0+ C_1/2 +C_2/3+.........+C_n/(n+1)= (2^(n+1)-1)/(n+1) .

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 =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove that : C_0+ 2C_1 +.........+2""^nC_n=3^n .

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 , 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) , then the sum C_(0) + (C_(0)+C_(1))+…+(C_(0) +C_(1) +…+C_(n -1)) is equal to

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : C_0-C_1/2+C_2/3-.....+(-1)^n C_n/(n+1)= 1/(n+1) .

If (1+x)^n =C_0+C_1 x+ C_2 x^2 +....... C_nx^n prove the following : C_0-2C_1+3C_2-.........+(-1)^n (n+1)C_n=0 .