s(n) means the sum of the first n natural numbers. Try inserting n = 8 or n = 100.
I think what bothers you is a variable. We could make the exact same argument just for e.g. n=100, but the proof would not require any of the characteristics of the number 100. All the used rules required only for it to be a natural number. So we generalize to get a more useful result and say n, meaning any natural number.
Strictly mathematically, you are right though. Ellipses are not a good practice and formal mathematicians avoid them. I should have used the formal notation, but then the post would target a more mathematical audience. And they already know this stuff.
Regarding teachers, I guess the solution is to ask a lot of questions. But that can be hard if you are in a class with people who don't respect interest in school subjects. I think that is a larger problem.