Prove that `((n+1)/(2))^(n) gt (n!)`

Prove that (C_(1))/(2)+(C_(3))/(4) +(C_(5))/(6)+….=2^(n)/(n+1) 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 ""^(2n+1)P_(n-1)=((2n+1)!)/((n+2)!) and ""^(2n-1)P_n=((2n-1)!)/((n-1)!)

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

If one root of the quadratic equation ax^(2)+bx+c= 0 is equal to the nth power of the order, then prove that (ac^(n))^((1)/(n+1)) + (a^(n)c)^((1)/(n+1))=-b

For all nge1 , prove that p(n):n^3+(n+1)^3+(n+2)^3 is divisible by 9.

Prove that 1(1) !+2(2) !+3(3) !+............ .+n(n) ! = (n+1) !-1

Prove that 1+(1)/(4)+(1)/(9)+...........+(1)/(n^(2))lt2-(1)/n,(ngt1)

Using mathematical induction prove that (2n+7) lt (n+3)^2 for all n in N

For all nge1 , prove that p(n):2^(3n)-1 is divisible by 7.