M1/M2 Mathematical Induction

1. Prove, by mathematical induction, that n³ - n + 3 is divisible by 3 for all positive integers n.

2. Prove that if n is an odd integer, x^n + 1 is divisible by x + 1.

3. Let a =/ 0 and a =/ 1. Prove by mathematical induction that [1/(a-1)] - (1/a) - [1/(a²)] -...- [1/(a^n)] = {1/[(a^n)(a-1)]} for all positive integers n.