In this chapter we explored sequences and mathematical induction. At first these might not seem entirely related, but there is a link: recursive reasoning. When we have many cases (maybe infinitely many), it is often easier to describe a particular case by saying how it relates to other cases, instead of describing it absolutely. For sequences, we can describe the
th term in the sequence by saying how it is related to the
previous term. When showing a statement involving the variable
is true for all values of
we can describe why the case for
is true on the basis of why the case for
is true.
While thinking of problems recursively is often easier than thinking of them absolutely (at least after you get used to thinking in this way), our ultimate goal is to move beyond this recursive description. For sequences, we want to find
closed formulas for the
th term of the sequence. For proofs, we want to know the statement is actually true for a particular
(not only under the assumption that the statement is true for the previous value of
). In this chapter we saw some methods for moving from recursive descriptions to absolute descriptions.
Throughout the chapter we tried to understand
why these facts listed above are true. In part, that is what proofs, by induction or not, attempt to accomplish: they explain why mathematical truths are in fact truths. As we develop our ability to reason about mathematics, it is a good idea to make sure that the methods of our reasoning are sound. The branch of mathematics that deals with deciding whether reasoning is good or not is
mathematical logic, the subject of the next chapter.