Hey there!

Not sure what you mean by 'normal' - if you could explain on this a bit more I might be able to give you more of an accurate answer, but I'll do the best I can

This is a complex question, and it does take a lot of foresight and thinking ahead to simplify; I doubt this question would appear as a standalone in an exam (it should appear with previous parts to simplify the question, ie. part a) prove the first result, b) hence prove by induction the second part) (and from what I've seen, it hasn't!). This is after all just a tough textbook question, so if you're looking to advance your skills and whatnot, this is definitely the sort of level of question you'd be looking to be able to solve at some point down the line.

I think I can assume that you're implying that the notation you don't understand is \(\in \mathbb{Z}^+\); the first symbol just means 'is a member of', the second means the set of positive integers. Combined, it just means such that n is a member of the set of positive integers. You definitely don't have to know this (you can write out where n is a positive integer every single time), but I choose not to, because it's extra characters that I don't need

If there's any other notation you don't understand, let us know and we'll gladly help you out

Hope this helps