Sum of Squares Formula Induction Proof

Prove that the sum of the squares of the first $n$ natural numbers can be expressed as: $1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$ for all natural numbers $n \geq 1$ using proof by induction.

This problem focuses on the fascinating technique of mathematical induction, which is a powerful tool used to prove statements about natural numbers. At its core, induction allows us to establish that if a statement holds for a particular case, and assuming it holds for an arbitrary case implies it holds for the next, then it must hold for all natural numbers beyond a base case. This is analogous to a domino effect, where knocking over the first domino causes all subsequent dominoes to fall.

In this case, we are tasked with showing that a specific formula describes the sum of the squares of the first n natural numbers. The formula is a compact representation of what would otherwise be a repetitive calculation if evaluated directly. The expression n times the quantity of n plus one times two n plus one, all divided by six, efficiently summarizes the result of the sum from one squared up to n squared.

During the process of solving such a problem by induction, you'll typically start by proving the base case, often n equals one, is true. Then, assuming the formula holds for some arbitrary value k, you'll demonstrate that it must also be valid for k plus one. This step, known as the inductive step, is crucial. It often involves algebraic manipulation and leveraging the assumption to prove the property for the next integer. Understanding induction well opens a door to proving many other properties and sequences within discrete mathematics, making it an essential topic in this field.