Logic

A Few Inductive Fibonacci Proofs

Having studied proof by induction and met the Fibonacci sequence, it’s time to do a few proofs of facts about the sequence. We’ll see three quite different kinds of facts, and five different proofs, most of them by induction. We’ll also see repeatedly that the statement of the problem may need correction or clarification, so …

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Introducing the Fibonacci Sequence

We’ve been examining inductive proof in preparation for the Fibonacci sequence, which is a playground for induction. Here we’ll introduce the sequence, and then prove the formula for the nth term using two different methods, using induction in a way we haven’t seen before.

Inductive Proofs: More Examples

Last week we looked at examples of induction proofs: some sums of series and a couple divisibility proofs. This time, I want to do a couple inequality proofs, and a couple more series, in part to show more of the variety of ways the details of an inductive proof can be handled.

Inductive Proofs: Four Examples

Last week we looked at introductory explanations of what mathematical induction is, including answers to some misunderstandings of the concept. But we only looked at one trivial example of such a proof; for a real understanding of the technique, we need some fuller examples. For that purpose, I have chosen a few questions we have …

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What is Mathematical Induction?

Last week’s exploration of a problem involving the Fibonacci sequence, has led me to delve into that and related concepts. In order to say much about the Fibonacci numbers, we have to first explore the concept of proof by mathematical induction. We’ll introduce it here, and then dig deeper next time.

More Handshake Problems

Last week we looked at problems about counting diagonals in a polygon, and the very similar problem of counting handshakes when everyone in a group shakes with everyone else. In the course of searching for those problems, I also found some very different problems that are also about handshakes. We’ll look at those here, just …

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