prove or disprove: the sum of 3 plus the square of any odd number is even

Answer 1

Hey Cesar,

It helps me to think about these problems if I write the problem more mathematically. In this case we want to prove that 3+odd2=even. By definition we know that an odd number is of the form 2n+1, n∈Z, and an even number is of the form 2m, m∈Z. If we plug this in we get that:

3 + (2n+1)² = 2m

3 + 4n² + 4n + 1 = 2m

4n² + 4n + 4 = 2m

2(2n² + 2n + 2) = 2m

m = 2n² + 2n + 2

This statement is true because integers are closed under addition and multiplication (meaning you can multiply n and add to it as much as you want and it stays an integer). Therefore the predicate is true. Hope this helps.

-Nathan