Introduction

Introduction

Francesco Maurolico used induction to prove that the sum of the first $n$ odd positives is $n^2$. A single counterexample can refute a claim (e.g. Euler's $n^2 - n + 41$ fails at $n = 41$). Pattern-spotting is not a proof; induction is.

Sums of odd numbers suggest $1 + 3 + \cdots + (2n-1) = n^2$, but infinitely many checks are impossible — induction handles this.