Mathematical Induction and the Binomial Theorem - Mathematics (Undergraduate Foundation)

This course covers two critical topics in discrete mathematics: the principle of mathematical induction as a formal proof technique, and the binomial theorem for expanding powers of binomials. The course structure directly links the proof method to one of its most important applications in algebra. These topics are the foundation of algorithm verification, probability theory, and statistics. Mathematical induction is a core technique for proving algorithm correctness and establishing formulas for series. The binomial theorem is a powerful tool used directly in statistics for the binomial distribution and in financial mathematics. By the end of this course, you will be able to apply the principle of mathematical induction to prove mathematical statements, solve complex counting problems using permutations and combinations, and use the binomial theorem to expand any binomial expression for a positive integer exponent. This course is for first-year university students and is a fundamental prerequisite for computer science, statistics, and probability theory. It provides essential skills for competitive mathematics and advanced algebraic studies.

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 establishes the essential groundwork for the course. It introduces the core logical principles and tools—induction, counting, and factorials—that are required to master the more advanced topics that follow. Key concepts covered are the logic of inductive reasoning, the fundamental principle for counting outcomes, and the definition and use of factorial notation.

Concept Overviews

4 Lessons

2. Applying Mathematical Induction

This chapter moves from the logic of induction to its formal application. It details the rigorous two-step process for constructing a valid inductive proof, a core skill in formal mathematics. Key topics include the structure of an inductive proof. Worked examples demonstrate the application to summation formulas, divisibility statements, and inequalities.

Concept Overviews

1 Lesson

Problem Walkthroughs

4 Lessons

3. Permutations and Combinations

This chapter covers combinatorics, the mathematics of counting. A mastery of this topic is the absolute prerequisite for understanding the coefficients in the binomial theorem and for any study of probability. Key topics include the distinction between permutations (arrangements) and combinations (selections), their respective formulas, and their application to problem-solving.

Concept Overviews

2 Lessons

Problem Walkthroughs

4 Lessons

4. The Binomial Theorem

This chapter provides a complete treatment of the binomial theorem. It presents the formula for expanding binomial expressions of the form (a+b)ⁿ for any positive integer n, a fundamental tool in algebra and statistics. Key topics include the statement of the theorem, the role of Pascal's triangle in finding coefficients, and methods for finding a specific term or coefficient in an expansion.

Concept Overviews

2 Lessons

Problem Walkthroughs

4 Lessons

5. Conclusion

This chapter consolidates the concepts of mathematical induction and the binomial theorem. It summarises the proof technique and the expansion formula, reinforcing their connection and importance. The conclusion summarises the two-step process of induction and the binomial formula. It also outlines how these concepts are directly applied in probability and statistics.

Concept Overviews

2 Lessons