# Monthly Archives: September 2016

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## Fibonacci Numbers and the Euclidean Algorithm

By |2017-01-01T15:39:00-08:00September 30th, 2016|Categories: Courses, Math 341||

As several of you noted in class Thursday, the Fibonacci numbers made a surprise appearance during an otherwise routine calculation of the greatest common divisor of two integers. Their appearance was not a coincidence. Let's take a look. The Fibonacci Numbers If you're interested in reading about Fibonacci and the Fibonacci numbers in detail, I encourage [...]

## A Tale of Two Proofs

By |2017-01-01T15:40:08-08:00September 23rd, 2016|Categories: Courses, Math 341|Tags: |

Today in class we saw one of the more fundamental properties of the binomial numbers, namely Pascal's Relation, which is the following: Pascal's Relation. For every $n\geq 0$ and $0\leq k\leq n$, we have the equality $\binom{n}{k}+\binom{n}{k-1}=\binom{n+1}{k}.$ We saw how this relation could be used to quickly compute binomial numbers and fill out Pascal's [...]

## Example: A Proof by Induction

By |2017-01-01T15:25:37-08:00September 22nd, 2016|Categories: Courses, Math 341|Tags: |

You may comfortable with the logic of mathematical induction, but still feel uncomfortable with how to actually write up the proof. So here is an example taken from class today, presented in several different ways. Let's prove the following: Claim. For every integer $n\geq 1$ the following equality holds: $$\sum_{i=1}^n (2i-1)= n^2.$$ In other words, we [...]

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#### This Is A Custom Widget

This Sliding Bar can be switched on or off in theme options, and can take any widget you throw at it or even fill it with your custom HTML Code. Its perfect for grabbing the attention of your viewers. Choose between 1, 2, 3 or 4 columns, set the background color, widget divider color, activate transparency, a top border or fully disable it on desktop and mobile.