Mathematical Induction - Mathematics (Undergraduate Foundation)

Mathematical induction proves a statement is true for every whole number using a starting point and a logical link. It works like a chain reaction where one step confirms the next. This course explains the core principle and applies it to many types of problems. You will study standard series, fractional series, divisibility, and inequalities. The content also covers recursive sequences, matrix powers, and set theory to give you a complete understanding of the method. Programmers use this logic to check that software code works without mistakes. Engineers use it to prove that formulas for structural designs or machines are safe. This skill is vital for anyone in technology, finance, or data science because it makes logical thinking a clear, repeatable process. It removes guesswork and provides the certainty needed for professional work in any technical field. You will learn the three-step induction method to solve different kinds of mathematical proofs. You will prove formulas for series, check if expressions are divisible, and handle difficult inequalities. You will also find general formulas for repeating sequences and solve problems involving matrices and sets. By the end, you will write clear, logical proofs that are required for university mathematics examinations. This course is for university students starting science or engineering degrees and senior secondary school students preparing for higher education. It is also helpful for workers who want to improve their reasoning and problem-solving. Even those not studying mathematics will benefit from the clear way of thinking this course teaches. It provides the basic skills needed for any career that requires logic and proof.

7 hrs

Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science. This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions. Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Course Chapters

1. Introduction

This chapter defines the principle of mathematical induction and its role in formal logic. You will understand the domino effect analogy and why this method is necessary for proving statements across infinite sets of integers. It provides the essential foundation for all subsequent proof techniques in this course. You will master defining the principle of induction, verifying the base case for n equals one, formulating the inductive hypothesis for n equals k, and executing the inductive step to prove the statement for n equals k plus one.

Concept Overviews

2 Lessons

45:31

2. Series

This chapter focuses on using induction to prove formulas for the sum of arithmetic and geometric sequences. It is the most common application of the method and essential for verifying that a general summation formula holds true for any number of terms. By the end, you will master proving linear and quadratic series identities, applying the inductive hypothesis to algebraic summations, and simplifying complex fractional series expressions to reach the required right-hand side.

Concept Overviews

1 Lesson

17:39

Problem Walkthroughs

3 Lessons

54:49

3. Fractional Series

This chapter covers proofs for the sums of series with terms written as fractions. It is a vital section because it forces you to use advanced algebraic manipulation and common denominators to link the inductive hypothesis to the final goal. By the end, you will master handling terms with product denominators, using common factors to simplify fractional expressions, and proving telescoping series identities through the inductive step.

Concept Overviews

1 Lesson

6:39

Problem Walkthroughs

2 Lessons

39:22

4. Divisibility

This chapter teaches how to use induction to prove that algebraic expressions are exactly divisible by specific integers. It is a critical skill for number theory and cryptography, moving beyond simple addition to master the manipulation of factors and remainders in a logical sequence. You will master expressing divisibility as a linear equation, manipulating exponents to isolate common factors, and applying the inductive hypothesis to prove that expressions like n cubed plus 2n are multiples of three.

Concept Overviews

1 Lesson

13:42

Problem Walkthroughs

3 Lessons

39:17

5. Inequalities

This chapter covers using induction to prove mathematical inequalities where one expression is always greater than another. It is a vital section because it requires logical estimation and transitivity rather than simple equality. This skill is essential for analyzing algorithm efficiency and bounding errors in numerical analysis. You will master handling base cases where n is greater than one, applying the inductive hypothesis to inequalities, and using algebraic estimation to prove that exponential growth eventually exceeds polynomial growth.

Concept Overviews

1 Lesson

9:31

Problem Walkthroughs

2 Lessons

27:18

6. Recursive Sequences

This chapter applies induction to sequences where each term depends on those before it. It is critical for computer science and engineering because it provides the mathematical proof for how recursive algorithms and loops function. You will learn to verify that a general formula correctly predicts the value of any term in a repeating sequence. You will master defining terms using recurrence relations, proving explicit formulas for recursive sequences, and applying the inductive step to link successive terms.

Concept Overviews

1 Lesson

15:11

Problem Walkthroughs

3 Lessons

44:47

7. Other Problems

This chapter covers miscellaneous induction proofs that do not fit into standard categories. It tests your ability to apply logical steps to unfamiliar mathematical structures, ensuring you can use the base case and inductive step on any technical problem. By the end, you will master proving formulas for the nth power of a matrix, calculating total subsets for a set with n elements, and applying inductive logic to non-standard algebraic identities.

Problem Walkthroughs

2 Lessons

31:54