# Proofs

## 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 …

## Graph Coloring: Working Through a Proof

(A new question of the week) The Math Doctors have different levels of knowledge in various fields; I myself tend to focus on topics through calculus, which I know best, and leave the higher-level questions to others who are more recently familiar with them. But sometimes, both here and in my tutoring at a community …

## 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.

## Multiplying Vectors II: The Vector Product

Last time, we looked at the scalar, or dot, product of vectors, focusing on proving the equivalence of two ways to define it. This time, we’ll look at the vector, or cross, product in the same way. The distinction between dot and cross product reflects the symbol used, u · v vs. u × v, …

## Multiplying Vectors I: The Scalar Product

Having covered the basics of defining and adding vectors, multiplying by scalars and finding unit vectors, it’s time to look at multiplying vectors together. What makes this entirely unlike working with numbers is that there are two ways (in fact, more than two!) to multiply two vectors. We’ll look at one of those today, the …