Let S U M S Q(k) denote the sum of the squares of the...

Let $S U M S Q(k)$ denote the sum of the squares of the natural numbers from 1 through $\mathrm{k}$, inclusively. Write a recursive definition of $S U M S Q(k)$ and prove that $S U M S Q(k)=(k(k+1)(2 k+1)) / 6$.

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Key Concepts

Recursive Definitions

A recursive definition specifies an object, such as a sequence or function, in terms of simpler instances of itself. It typically includes a base case that provides the starting value, and a recursive step that defines each subsequent element based on previous element(s). This concept underpins many processes in mathematics and computer science where a problem is broken down into simpler, more manageable subproblems.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a property holds for all natural numbers. It involves two main steps: first, verifying the property for a base case (usually the smallest natural number), and second, proving that if the property holds for an arbitrary natural number k, then it must also hold for k+1. This method is essential for proving formulas and statements defined recursively.

Sum of Squares Formula

The sum of squares formula provides a direct, closed-form expression for calculating the sum of the squares of the first k natural numbers. The formula, k(k+1)(2k+1)/6, is derived either through algebraic manipulation or induction and eliminates the need to compute each square individually, thus offering an efficient means to compute the sum.

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