# Mathematics: Proofs by induction

Mathematics from high school to university
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How to conduct proofs by induction and in what circumstances we should use them.
Prove (by induction) some formulas holding for natural numbers.
Prove (by induction) some statements about divisibility of natural numbers.
Prove (by induction) explicit formulas for sequences defined in a recursive way.
Prove (by induction) some simple inequalities holding for natural numbers.
You will also get an information about more advanced examples of proofs by induction.
You will get a short explanation how to use the symbols Sigma and Pi for sums and products.

## Requirements

• Being able to perform arithmetical operations (addition, subtraction, multiplication, division) on real numbers, in particular fractions.
• Being familiar with basic rules for computations with real numbers such as the commutativity law, the associativity law (for addition and multiplication) and the distributivity law (F.O.I.L. method / parenthesis multiplication; the square of a sum and of a difference).
• Being familiar with the rules for powers with natural exponents (the zero-power rule and the product rule).
• Being familiar with the concept of divisibility for natural numbers.
• Being able to handle variable expressions and understanding that all the regular rules for computations with real numbers also hold for algebraic expressions.
• Being interested in mathematics or at least feeling the need of studying and understanding it.

## Description

How would you prove that a theorem or a formula is true for *all* natural numbers? Try it for n=0, n=1, n=2, etc? It seems to be a lot of work, or even completely impossible, as there are infinitely many natural numbers!

Don't worry, there is a solution to this problem. This solution is called "proof by induction" and this is the subject of this short (and free) course. The Induction Principle is often compared to the "domino effect", which will be illustrated in the course. (This is also the reason for our course image.)

In this course you will learn how induction proofs work, when to apply them (and when not), and how to conduct them. You will get an illustration of this method on a variety of examples: some formulas, some inequalities, some statements about divisibility of natural numbers. You will also get some information about other courses where you can see some theory, and more advanced proofs based on the same principle.

Sadly, there is no possibility of asking question in free courses, but you can ask me questions about this subject via the QA function in my other course: "Precalculus 1: Basis notions", where the topic of proofs by induction is covered, both theoretically (Peano's axioms) and practically, with several examples.

## Who this course is for:

• High school students who want to learn conducting proofs by induction.
• University or college students who have discovered that they need to master proofs by induction for some university level courses, and they want to re-learn this method of proving theorems.
• Everybody who likes mathematics and want to learn more of it.

## Instructors

University teacher in mathematics, PhD

I am a multilingual mathematician with a passion for mathematics education. I always try to find the simplest possible explanations for mathematical concepts and theories, with illustrations whenever possible, and with geometrical motivations.

I worked as a senior lecturer in mathematics at Uppsala University (from August 2017 to August 2019) and at Mälardalen University (from August 2019 to May 2021) in Sweden, but I terminated my permanent employment to be able to create courses for Udemy full-time.

I am originally from Poland where I studied theoretical mathematics and got pedagogical qualifications at the Copernicus University in Torun (1992-1997). Before that, I enjoyed a very rigorous mathematical education in a mathematical class in high school "Liceum IV" in Torun, which gave me a very solid foundation for everything else I have learned and taught later.

My PhD thesis (2009) was at Uppsala University in Sweden, with the title: "Digital Lines, Sturmian Words, and Continued Fractions".

In 2018 I received four pedagogical prizes from students at the Faculty of Science and Technology of Uppsala University: on May 13th from the students at the Master Program in Engineering Physics; on May 25th from the students at the Master Program in Electrical Engineering; on December 20th from the students at the Master Program in Chemical Engineering; on January 10th 2019 from UTN (Uppsala Union of Engineering and Science Students at Uppsala University).

I speak Polish, Swedish, English, Dutch, and some Russian; learning Ukrainian.

Editor at MITM AB

I have a background in medicine and software development. I've done enough mathematics to at least follow along in Hania's courses and I'm learning a lot as I edit the material. I have also written a book about medical software design as it pertains to the medical record ("Rethinking the electronic healthcare record"). For Hania's math courses, it's my job to set up the environment and produce the final output that goes into these courses.