Prove that `(C_(1))/(2)+(C_(3))/(4) +(C_(5))/(6)+….=2^(n)/(n+1)` where `C_(r) =^(n)C_(r)`

Find the sum C_(0)-C_(2) +C_(4)-C_(6)+…. , where C_(r)=^(n)C_(r)

Prove that (""^(n)C_(0))/(1)+(""^(n)C_(2))/(3)+(""^(n)C_(4))/(5)+(""^(n)C_(6))/(7)+...=(2^(n))/(n+1)

Prove that ""^(n+1)C_(r+1)=(n+1)/(r+1)"^nC_r

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

Prove that ""^nC_r+^nC_(r-1)=^(n+1)C_r

Prove that C_0C_2+C_1C_3+…+C_(n-2)C_n=^(2n)C_(n-2)

Prove that C_0+C_2/3+C_4/5+...=(2^n)/(n+1)

Prove that (C_0+C_1)(C_1+C_2)(C_2+C_3)...(C_(n-1)+C_n)=(C_0C_1C_2...C_(n-1)(n+1)^n)/(n!)

Three distinct A.P. s' have same first term common differences as d_(1) ,d_(2) ,d_(3) and n^(th) terms as a_(n) , b_(n) ,c_(n) respectively such that (a_(1))/(d_(1))=(2b_(1))/(d_(2))=(3c_(1))/(d_(3)). If (a_(7))/(c_(6))=(3)/(7) then find the value of (b_(7))/(c_(6)) .