Math Project

1. Pick out the 4 or (arguably) 5 theorems that would be best proved using Mathematical Induction.
2. Rewrite theorems 1 and 6 as conditional statements.
3. Describe the strategy required to prove theorem 3.
4. List all axioms and/or definitions needed to complete these proofs.

Use the following theorems to answer each question above:

1)The sum of two even integers is even.

2)If n is an odd integer, then n/2= (1)2.

3)These statements about real numbers x are equivalent:

(i) x is rational.

(ii) x/2 is rational.

(iii) 3x-1 is rational.

4)Disprove: Every positive integer can be written as the sum of the squares of three integers.

5)13+ 23+ ... + n3= (n(n+1)/2)2for positive integers n.

6)The product of two odd numbers is odd.

7)3 divides n3+ 2nfor positive integers n.

8)For all integers n, if 3n+ 2 is even, then n is even.

9)12+14+18++12=212for positive integers n.

10)For integers m, n, and p, If m+ n and n+ pare even integers, then m+ p is even.

11)For any integer n, if n2is divisible by 4, then n is even.

12) 11! + 22! + ... + nn! = (n+ 1)! -1 for positive integers n.