How do you prove statements about infinitely many cases all at once? This series
covers the Principle of Mathematical Induction, Strong Induction and Recurrence
Relations, essential tools for reasoning about sequences and recursive
algorithms.
The articles are listed below, in the order they were written.
How do you prove a formula holds for every natural number, if there are infinitely many of them? With mathematical induction. This introduction maps out the series: classic induction, strong induction and recurrence relations, from Fibonacci to the Tower of Hanoi.
If the first domino falls and each domino knocks over the next, they all fall. That is the logic of the Principle of Mathematical Induction. We will see the structure of the proof (base case, hypothesis and step) in sums, divisibility and inequalities, and the false proof that all horses are the same color.
What if the step depends on more than one earlier case, as in Fibonacci? Strong induction assumes all the earlier cases. We will see the technique at work on prime factorization, the chocolate bar, postage stamps and Binet’s formula, which in floating point fails from the 71st term on.
How many moves does it take to solve the Tower of Hanoi? The answer comes from a recurrence relation, which writes the problem in terms of itself with one disk fewer. We set up recurrences and solve them by substitution, by the characteristic equation and by the Master Theorem, used for merge sort.
