Week 12-Proof by Induction

        This week, I was taught proof by induction. This topic is not unfamiliar to me since I already learn it from MAT137. However, by re-learning it in CSC165, my understanding was solidified.  I think the hard part about proof by induction is the idea that by assuming a term works and later proving that the next term works, I am able to demonstrate that all term works. After being taught again in CSC165, I was convinced by the prof that proof by induction does work.
         Proof by induction is a proof technique used to establish a given statement for all natural numbers. It is similar to the idea of dominoes, in which by knowing that domino0(d0) works, I know that d1 works, then d2, then d3 and so on until all dominoes (all natural numbers) have been enumerated. The general structure for proof by induction goes like this:
 1. Assume d0 does fall
 2. Base case: d0 falls # by assumption
 3. Induction step: - assume n a generic natural number, and that dn falls
                              - then dn+1 falls # dn will hit dn+1
                              - then for all n, dn falls implies dn+1 falls
                              - then for all n, dn falls
                              - then d0 falls implies for all n, dn falls
Basically, the central idea is that after proving that the base case works, assume the assumption works for dn, and try to arrive at a conclusion that dn+1 also works.
        Here is an example of proof by induction, (without the quantifiers and bookends):
 claim:

proof:

Lastly, proof by induction is a topic that comes easy to me since I already learned it before. However, the problem about diagonals presented in class troubles me. I was unable to figure out a clear pattern to the boxes since there are too many different combinations of dimensions for the box. Thanks to Loora's blog, I was able to gain some insight.

http://looralicsc165slog.blogspot.ca/2014/11/last-problem-solving.html?showComment=1417589892018#c163994337424380139