^nC_(0)-(1)/(2)^(n)C_(1)+(1)/(3)^(n)C_(2)-...+(-1)^(n)(^nC_(n))/(n+1)=

nC_(0)-(1)/(2)(^(^^)nC_(1))+(1)/(3)(^(^^)nC_(2))-....+(- 1)^(n)(nC_(n))/(n+1)=

(nC_(0))^(2)-(nC_(1))^(2)+(nC_(2))^(2)+....+(-1)^(n)(nC_(n))^(2)

Prove that ^nC_(0)^(n)C_(0)-^(n+1)C_(1)^(n)C_(1)+^(n+2)C_(2)^(n)C_(2)-...=(-1)^(n)

Prove,by induction,that (nC_(0))/(x)-(nC_(1))/(x+1)+(nC_(2))/(x+2)-.........+(-1)^(n)*(nC_(n))/(x+n)=(n!)/(x(x+1)(x+2)......(x+n)),x in

Prove that (nC_(0))/(x)-(n_(C_(0)))/(x+1)+(^nC_(1))/(x+2)-...+(-1)^(n)(n_(n))/(x+n)=(n!)/(x(x+1)...(x-n)) where n is any positive integer and x is not a negative integer.

Prove that (1)/(n+1)=(nC_(1))/(2)-(2(^(n)C_(2)))/(3)+(3(^(n)C_(3)))/(4)-...+(-1)^(n+1)(n(^(n)C_(n)))/(n+1)

Prove that ^nC_(0)^(2n)C_(n)-^(n)C_(1)^(2n-1)C_(n)+^(n)C_(2)xx^(2n-2)C_(n)++(-1)^(n)sim nC_(n)^(n)C_(n)=1

4^(n)C_(0)+4^(2)(nC_(1))/(3)+4^(3)(nC_(2))/(3)+4^(4)(nC_(3))/(4)+....+4^(n+1)(nC_(n))/(n+1) is equal to

Prove that: (1)/(2)nC_(1)-(2)/(3)nC_(2)+(3)/(4)nC_(3)-(4)/(5)nC_(4) +....+((-1)^(n+1)n)/(n+1)*nC_(n)=(1)/(n+1)