Prove the following theorem:
Theorem
∀n ∈ Z, n is either even or odd (but not both).
Your proof must address the following points:
1. n is even or odd (and nothing else).
2. n is odd =⇒ n is not even (hint: contradiction).
3. n is even=⇒ n is not odd (hint: contrapositive).
The first point is a bit more difficult. Start by making a
statement about 0. Then assuming that n is even, what can you say
about...
Prove using mathematical induction: 3.If n is a counting number
then 6 divides n^3 - n. 4.The sum of any three consecutive perfect
cubes is divisible by 9. 5.The sum of the first n perfect squares
is: n(n +1)(2n +1)/ 6