Mathematics: Proofs by induction

Name: Mathematics: Proofs by induction
Rating: 4.9 (197 reviews)

Mathematics from high school to university

Created byHania Uscka-Wehlou, Martin Wehlou

Last updated 5/2026

English

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What you'll learn

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.

Course content

3 sections • 13 lectures • 1h 57m total length

Introduction to the course4:17
Explore the fundamentals of mathematical induction and how proofs by induction work through concise examples in this free two-hour course. Access notes and PDFs to reinforce definitions and computations.
What kinds of statements can be proven by induction6:45
Learn which statements can be proven by induction, whether they hold for all natural numbers or from a starting point, and connect to the domino effect and t(n) notation.
Induction: this is how it works27:03
Explore the principle of mathematical induction, establish base cases and the induction step, and prove the sum of the first n natural numbers using t(n) and sigma notation.
Both cases are necessary7:29
Demonstrate the necessity of both the base case and the induction step in proofs by induction, with two examples showing why each part is essential.

Proving formulas, Problem 10:06
Sequences: guess and prove, Problem 210:04
Guess and prove an explicit formula for a recursive sequence, deriving a_n = 2^n from a_0 = 1 and a_{n+1} = 2 a_n, and verify by induction.
Sequences: guess and prove, Problem 3 with two base cases16:36
Guess and prove the closed formula for a sequence defined by a1=3, a2=7 and a_n=3 a_{n-1}-2 a_{n-2} for n>=3. Prove that a_n=2^{n+1}-1 by induction with two base cases.
Proving divisibility, Problem 411:20
Learn how to prove divisibility by induction: show that n^3 minus n is divisible by six (and by nine) for all natural numbers, using base cases and inductive steps.
Not necessarily for all natural numbers: an inequality, Problem 56:29
Explore induction to prove a strict inequality for natural numbers, starting from the base case n = 3 and proving n + 1 using the induction hypothesis and power-of-two expressions.
A difficult proof, Problem 611:40
Apply induction to prove the formula for all real a, b and positive integers n, using a base case, a sigma notation view, and a t(n) to t(n+1) transition.
Another difficult proof, Problem 74:39
Rewrite the left-hand side as a product ∏_{k=1}^n (1 - 4/(2k−1)^2), then extend to n+1 and use pi notation to guide problem 7 in this induction-based proof.
Proofs by induction, Wrap-up6:38
Explore proofs by induction across natural numbers, divisibility, and matrix or polynomial contexts, compare induction to the domino effect, including strong induction and when induction is optimal.

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

[None of our courses are produced using AI; they are all real-human products.]

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.

Mathematics: Proofs by induction

What you'll learn

Explore related topics

Course content

What is mathematical induction and how it works4 lectures • 46min

Examples of proofs by induction8 lectures • 1hr 8min

Extras1 lecture • 5min

Requirements

Description

Who this course is for: