Algebra

The Golden Ratio and Fibonacci

We’re looking at the Fibonacci sequence, and have seen connections to a number called phi (φ or \(\phi\)), commonly called the Golden Ratio. I want to look at some geometrical connections and other interesting facts about this number before we get back to the Fibonacci numbers themselves and some inductive proofs involving them.

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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Disappearing Area?

We’ve been looking at dissection puzzles, where we cut an object into pieces, and rearrange them. Here we’ll examine a mystery posed by two different puzzles, each of which seems to change the area by rearranging the pieces. The answer combines the marvelous Fibonacci numbers and [spoiler alert!] how easily we misjudge areas.

A Geometrical Limit

(A new question of the week) We usually see limits applied to functions in a calculus class. An interesting question from late October deals with a limit in a geometrical construction based on a function. We’ll be seeing how to discover a proof, then several alternative proofs, and finally what the answer means.

A Proof Problem: Chords and Tangents

One thing we enjoy doing is guiding a student through the process of problem-solving. Here is a problem from August that illustrates how to think through a complicated geometrical proof. In particular, this uses some circle theorems involving chords, secants, and tangents, together with a bit of algebra.

A Finite Series Workout

(A new question of the week) A question from the end of August led a student and a Math Doctor to an extra challenge, by way of an apparent typo in the problem. We particularly enjoy working with students who are willing to take on extra work in order to learn more than they need …

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