I am a TA for a CS course and on an exam I was grading I had a "strong" discussion with the professor that the answer was incorrect involving strong induction. His explanation of strong induction is this:
Given some property P over the natural numbers we want to prove P(n) is true.
1. Let b be the base case and prove P(b) is true.
2. The inductive hypothesis is to assume that for all i, b < i < n, P(i) is true.
I tried to argue that if you don't assume that P(i) is true when i equals b then you can't use it. He says it is pointless to assume it is true because you already proved it in step 1. I pointed out that multiple books state that they include b but he just waved it off and said they were sloppy.
Does it matter if you include the base case or not in the inductive hypothesis?
Practically speaking, it doesn't matter much. You're allowed to use the fact that $P(b)$ is True whether or not you assume it, because you've proven it. Your professor is right that you don't have to assume it, and can only assume $P(i)$ for $b<i<n$. In that sense, it's more precise to not assume it for $b$. Personally I don't view the difference as particularly important and wouldn't have taken any points off.