Consider the theorem: "If n is even, then n + 1 is odd."

• This is a really "out-there" way to prove what is a very straightforward direct proof.
• I haven't had a lot of success asking groups to just prove the theorem using induction. That's partially because it's the last technique in the world you would normally attempt on a theorem like this, and partially because it isn't a summation formula like so many of our examples were. But, there is still value in considering it, so...
• I want you to tackle a few questions this week that would get you thinking about how to set this up.

Let P(n) be "for all positive integers n, the integer immediately following the n^th positive even integer is odd." Answer the following:

1. What is the benefit of writing P(n) in terms of the n^th positive even integer instead of working with simply an n (even) and n+1 (odd) when we are attempting induction?
2. What would the Basis Step look like? What is the first n value: 1 or 2?
3. What do we assume to be true in the Inductive Step?
4. What are we trying to show to be true in the Inductive Step?
5. How is this different from the way we use Induction to prove Summation Formulae?