Induction by Contradiction

So for years I used Proof By Induction, but never really understood why it worked. This frustrated me, and so I set out to discover the «proof» for proof by induction. I searched far and wide in all my textbooks and just kept finding the domino analogy to justify the three steps. Sure the analogy is memorable, but to me it never seemed like a proof. So after looking up induction in nearly every book I have, I found a decent explanation in Paul Foerster's Precalculus. He uses Proof By Contradiction to develop induction and the method is both clear and logical. Unfortunately this great induction lesson has been relegated to an appendix, and there are no exercises at all (particularly unfortunate since Foerster's claim to fame is his problem sets). I used Paul's explanation to create a lesson, along with problems, and I have attached both below. I'm not teaching induction this year, but induction came up in a conversation I was having with my friend Bryan who is. The worksheet below is updated from the first time I taught this four or fives years ago. The last two times I taught induction, however, I turned the worksheet into HW problems to fit my Exeter style problem sets, these, and a whole strand of induction problems I used in subsequent problem sets are included here.