CS350, Homework for Objective #5, Perform inductive proofs.


Prove that for every n > 3, 2^n < n!, where ^ denotes exponentiation.

Prove that the sum of the first n odd natural numbers is n^2.

Prove that for every natural n, n^3-n is a multiple of 3.

Prove that for every natural n, 5^n-1 is a multiple of 4.


Hints: 
These steps are useful for a proof.
1. Compute a few instance of the formula you are asked to prove.
2. State and prove the base case.
3. State the induction hypothesis.
4. Prove the induction case.
5. Show where the induction hypothesis is applied.