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)) +….(n+1)` terms = 0

If ""^(n)C_(r) +""^(n) C_(r+1)= ""^((n+1)) C_(x) then x=

Prove that C_(0)-(1)/(3)*C_(1)+(1)/(5)*C_(2) - …+(-1)^(n)*(1)/(2n+1)C_(n) =(2^(2n)(n !)^(2))/((2n+1)!)

"Prove that "(""^(4n)C_(2n))/(""^(2n)C_(n))=(1.3,5......(4n-1))/({1.3.5....(2n-1)}^(2))

Show that (n !)/(x(x+1)(x+2)...(x+n))=(""^(n)C_(0))/(x)-(""^(n)C_(1))/(x+1) + (""^(n)C_(2))/(x+2)-(""^(n)C_(3))/(x+3) + ...+ ((-1)^(n)""^(n)C_(n))/(x+n) .

Use the identity (1+x)^(m)(1+x)^(n)=(1+x)^(m+n) to prove Vandermonde's theorem, ""^(m)C_(r ) +""^(m)C_(r-1). ""^(n)C_( 1)+""^(m)C_(r -2) *""^(n)C_(2) +….+""^(n)C_(r )=""^((m+n))C_(r )

Prove that sum_(r = 1)^n r^3 ((C_r)/(C_(r - 1))^2) = (n (n + 1)^2 (n+2))/(12)

Prove that following C_(0)+(3)/(2).C_(1)+(9)/(3).C_(2)+(27)/(4).C_(3)+……+(3^(n))/(n+1).C_(n)=(4^(n+1)-1)/(3(n+1)).