I really did not understand the significance of mathematical induction. It seems to me like for a given sentence you prove that it works for an actual case, then for a generalized case and then for another generalized case proving that it works the same way. I can see that it could be useful in cases lie that shown in the book about series and sums. I just do not yet have a clear picture on how it would work.
I was excited by the example of counting squares. I have had that problem given to me in the past but I just counted them out. I am excited now though because I see the generalized case where a square is dimensions n x n, the number of squares is n^2 + (n-1)^2 + (n-2)^2+...+(n-k)^2, where n = k.
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