Prove that `1-^n C_1(1+x)/(1+n x)+^n C_2(1+2x)/((1+n x)^2)-^n C_3(1+3x)/((1+n x)^3)+`

Prove that 1-^(n)C_(1)(1+x)/(1+nx)+^(n)C_(2)(1+2x)/((1+nx)^(2))-^(n)C_(3)(1+3x)/((1+nx)^(3))+

Prove that (^n C_0)/x-(^n C_0)/(x+1)+(^n C_1)/(x+2)-+(-1)^n(^n C_n)/(x+n)=(n !)/(x(x+1)(x-n)), where n is any positive integer and x is not a negative integer.

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) .

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_0)/x-(""^n C_1)/(x+1)+(""^n C_2)/(x+2)-.....+(-1)^n(""^n C_n)/(x+n)=(n !)/(x(x+1) . . . (x-n)), where n is any positive integer and x is not a negative integer.

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)^(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 .

The value of ""(n)C_(1). X(1 - x )^(n-1) + 2 . ""^(n)C_(2) x^(2) (1 - x)^(n-2) + 3. ""^(n)C_(3) x^(3) (1 - x)^(n-3) + ….+ n ""^(n)C_(n) x^(n) , n in N is

The value of ""(n)C_(1). X(1 - x )^(n-1) + 2 . ""^(n)C_(2) x^(2) (1 - x)^(n-2) + 3. ""^(n)C_(3) x^(3) (1 - x)^(n-3) + ….+ n ""^(n)C_(n) x^(n) , n in N is

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