Mathematical Induction (With Lots of Examples) [Video]

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Dave4Math » Number Theory » Mathematical Induction (With Lots of Examples) [Video]

I work through several examples of writing a proof by Mathematical Induction (for beginners). I concentrate on cases that demonstrate how to use mathematical induction to prove a statement true for all natural numbers. Afterward, I discuss Strong Induction and show how to use it. Then well-known arithmetic and geometric progressions formulas are proven using induction. Towards the end, I confirm that the Well-Ordering Axiom, Mathematical Induction, and Strong Induction are all logically equivalent.

In this video, I discuss a powerful method called mathematical induction. Hi, I’m Dave, welcome back to my channel. In this episode, firstly, I’m going to work through several examples that demonstrate how to use induction. I’ll also talk about the Well-Ordering Axiom and Strong Induction. After that, I work through several more examples, I prove that the Well-Ordering Axiom, Induction, and Strong Induction are all logically equivalent.

Before you continue, check out the video notes here on mathematical induction. You may also be interested to know that this video is part of the Number Theory Series playlist.

Mathematical Induction (With Lots of Examples)

This video is for anyone looking for a how-to learn Mathematical Induction, Strong Induction, and the Well-ordering Axiom.

How-to Learn Mathematical Induction

Induction is challenging, especially for beginners. That’s why I made this video so you can become skilled.

Step 1: Read what mathematical induction is.

Firstly, for your first reading of the induction statement, make sure that you understand every symbol and word in the statement.

Step 2: Work out some easy induction examples.

Secondly, here are two easy Mathematical Induction examples I work through in the video.

Step 3: Work out some intermediate examples.

Thirdly, these types of examples often involve an inequality or a factorial. 

Step 4: Read and understand what Strong Induction is.

After that, your first reading of Strong Induction is to familiarize yourself with the statement.

Step 5: Work out some Strong Induction examples.

After that, make sure that you understand the difference between induction and Strong Induction.

Step 6: Understand what the Well-Ordering Axiom is.

After that, I go over the Well-Ordering Axiom in the pdf notes in greater detail than in the video.

Step 7: Know that all three are equivalent to each other.

Finally, slowly and carefully go through the proof that all three are equivalent to each other.

What is an excellent mathematical induction example?

Mathematical induction allows us to prove statements that depend on a natural number to be true for all natural numbers. For anyone to accomplish an induction proof, a base case and an inductive step must be justified. 

Is mathematical induction challenging?

In general, learning how to write proofs is hard, especially induction. Proof by induction is typically hard to understand because of the inductive step, requiring a student to make a hypothesis, deduce a conclusion, and then conclude the proof.

Why do we use mathematical induction?

People use induction because it is potent. To be able to prove a result (with infinitely many cases) correct is compelling. Moreover, once someone practices induction, it becomes effortless to use.

How do you do strong induction proof?

Strong induction is very similar to induction; however, the inductive step is different. With mathematical induction, the inductive step is valid for a natural number implies valid for the next natural number. In proof by strong induction, the inductive step is the following: valid for the first consecutive finite number of cases implies true for the next natural number.

In conclusion, I want to turn it over to you. 

Does induction always have to have a base case?

So, either way, let us know what you think in the comments of this video: Mathematical Induction right now.

dave at dave4math

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.