Strong Form of Induction

The Strong Form of Induction (with Examples)

You’re trying to prove a statement is true using mathematical induction, but then you realize that the induction hypothesis doesn’t seem strong enough. What do you do? It turns out that there is more than one way to do induction. Let’s learn what strong induction is.

Arithmetic and Geometric Progressions

Arithmetic and Geometric Progressions

In this video, I am going to explain what arithmetic and geometric progressions are. I’ll also give several basics examples. Then I will prove the summations formulas for both the arithmetic and geometric progressions. In the end, I’ll discuss how these progressions arise in nature.

Induction Principles

Induction Principles (Basis Step Required)

In this video, I discuss various forms of mathematical induction, including strong induction. I establish that the base case of an induction argument is required. Also, I explain the differences between the second-order logic induction principle and the first-order statement.

Examples Using Mathematical Induction

Examples Using Mathematical Induction

In this video, I’m going to work through five examples using mathematical induction. The first two examples involve summations, and the last three examples have an inequality. In each of these examples, I work through specials cases before wiring a proof so that you can get a feel for the result before actually proving the result holds using mathematical induction.

Well Ordering Axiom

Well Ordering Axiom (Principle or Theorem?)

In this video, I’m going to talk about the Well-Ordering Axiom. First, I’ll discuss the ordering and why it’s reflexive, antisymmetric, and transitive. Then I explain the Well-Ordering Axiom followed by some basic examples. After that, I’ll discuss whether it should be an axiom or a theorem.

First Examples Using Mathematical Induction

First Examples Using Mathematical Induction

In this video, I’m going to cover three first examples using mathematical induction. So I’m going to assume that you have never seen mathematical induction before. So this is the first step, in a series of steps, to learn how to use and what mathematical induction is. Of course, to do that, we need to start with what the natural numbers are.

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Tonelli-Shanks Algorithm (by Example)

Okay, so you understand how to check if a quadratic congruence is solvable, but how do you find the solutions? In this article, I cover the Tonelli-Shanks algorithm by working through several examples. I also give a complete solution to a general quadratic congruence equation.

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Euler’s Totient Function and Euler’s Theorem

Many people have celebrated Euler’s Theorem, but its proof is much less traveled. In this article, I discuss many properties of Euler’s Totient function and reduced residue systems. As a result, the proof of Euler’s Theorem is more accessible. I also work through several examples of using Euler’s Theorem.

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Fermat’s Theorem (and Wilson’s Theorem)

Maybe you have heard of Wilson’s Theorem? But did you know that’s is converse also holds. In this article, I prove both Wilson’s Theorem, its converse, and Fermat’s Theorem. Then you will also see many examples using Fermat’s theorem.