Mathematical Induction by Titu Andreescu and Vlad Crisan is the definitive source and collection of elegant problems originated from various math competitions. Although the mathematical induction is one of the basic tools for all mathematicians and readers might feel familiar with, Titu’s book brings the method to a completely different level.
The first chapter lays down the fundamental of the method, by going through notations, variants, paradox and well-ordering etc. Chapter 2 shows the traditional area of induction in sums and product identities. The most inspiring chapters are from chapter 3 to chapter 9, where over ten different areas of mathematics are explained using induction method: functional equations, inequalities, sequences, number theory, combinatorics, games, geometry, calculus etc. If I didn’t read this book, I will never know that induction can be applied in so many areas. I feel I am fully charged after carefully working through this book and the understanding of induction method has never being better.
Reading a book like this is a serious commitment on your time and effort. Mathematics is better learned by thinking and solving problems. Examples and problems in each chapter are from important Olympiad competitions like IMO shortlist, IMO, USAMO, APMO, China CMO, St. Petersburg, TOT, and lots of author’s own. They are never short of fun and challenges. The best thing is that each problem has a full detailed solution at the end of the book, therefore making self-study possible. I highly recommend this book to all mathlets working towards national or international math Olympiad. I fully agree with the author’s statement that “the book can serve as a very good resource and teaching materials for anyone who wants to explore the beauty of Induction and its applications”.