Mathematical Formula

A possible way to solve this exercise is to use known mathematical formulas:

• The sum of positive integers up to n, inclusive, is n * (n + 1) / 2. Squaring that sum gives squareOfSum.
• The sum of the squares of positive integers up to n, inclusive, is n * (n + 1) * (2 * n + 1) / 6.

def sumFirstNPositive (n : Nat) : Nat :=
    n * (n + 1) / 2

def squareOfSum (number : Nat) : Nat :=
    (sumFirstNPositive number) ^ 2

def sumOfSquares (number : Nat) : Nat :=
    (number * (number + 1) * (2 * number + 1)) / 6

def differenceOfSquares (number : Nat) : Nat :=
    (squareOfSum number) - (sumOfSquares number)

This is probably the most efficient approach, since the problem reduces to a few constant multiplications and divisions, whereas the other approaches are O(n). On the other hand, the solution becomes more opaque because these formulas are not obvious from the structure of the problem. They require previous knowledge or further research.