If a is an integer greater than 1, show that `a^n-1` is divisible by `a -1` for all positive integers n.

Show that 9^(n+1) -8n -9 is divisible by 64, whenever n is a positive integer.

Show that 9^(n+1) - 8n -9 is divisible by 64, whenever n is a positive integer.

If n is any positive integer , show that 2^(3n +3) -7n - 8 is divisible by 49 .

Prove that (n!)^2 le n^n. (n!)<(2n)! for all positive integers n.

For every positive integer n, prove that 7^n-3^n is divisible by 4.

If p is a natural number, then prove that p^(n+1) + (p+1)^(2n-1) is divisible by p^(2) + p +1 for every positive integer n.

Using mathematical induction prove that d(x^n)/dx = nx^(n-1) for all positive integers n.

Integer just greater then (sqrt(3)+1)^(2n) is necessarily divisible by