# Induction

## Complex Numbers: Multiplication and Rotation

Having looked at the idea of complex numbers and how to perform basic operations on them, we are ready for one of the most important features for applications: their relationship to rotation. We’ll see this first in describing complex numbers by a length and an angle (polar form), then by discovering the meaning of multiplication …

## Fibonacci, Pascal, and Induction

A couple weeks ago, while looking at word problems involving the Fibonacci sequence, we saw two answers to the same problem, one involving Fibonacci and the other using combinations that formed an interesting pattern in Pascal’s Triangle. I promised a proof of the relationship, and it’s time to do that. And while we’re there, since …

## Generalizing and Summing the Fibonacci Sequence

Continuing our look at the Fibonacci sequence, we’ll extend the idea to “generalized Fibonacci sequences” (with different starting numbers), and see that the ratio of consecutive terms is the same in general as in the usual special case. Then we’ll look at the sum of terms of both the special and general sequence, turning it …

## 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 …

## 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 …

## 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.