Show that `(1+x)^(n) - nx-1` is divisible by `x^(2)` for `n gt N`

((n+2)!)/((n-1)!) is divisible by

show that [(sqrt(3) +1)^(2n)] + 1 is divisible by 2^(n+1) AA n in N , where [.] denote the greatest integer function .

Statement 1: 3^(2n+2)-8n-9 is divisible by 64 ,AAn in Ndot Statement 2: (1+x)^n-n x-1 is divisible by x^2,AAn in Ndot

Show that n^3+(n+1)^3+(n+2)^3 is divisible by 9 for every natural number n .

Statement 1: 2^(33)-1 is divisible by 7. Statement 2: x^(n)-a^(n) is divisible by x-a , for all n in NN and x ne a .

The condition that x^(n+1)-x^(n)+1 shall be divisible by x^(2)-x+1 is that

For each n in N, n(n+1) (2n+1) is divisible by

Prove the following by the principle of mathematical induction: \ x^(2n-1)+y^(2n-1) is divisible by x+y for all n in NNdot

Show that n^2-1 is divisible by 8, if n is an odd positive integer.

Statement -1 for all natural numbers n , 2.7^(n)+3.5^(n)-5 is divisible by 24. Statement -2 if f(x) is divisible by x, then f(x+1)-f(x) is divisible by x+1,forall x in N .