Neliösumma

Neliösumma is a Finnish mathematical term that refers to the sum of squares. It is used in various branches of mathematics to denote the quantity obtained by squaring each term and then adding the results. For a finite sequence x1, x2, ..., xn, the neliösumma is x1^2 + x2^2 + ... + xn^2, often written as Σ xi^2. In geometry and linear algebra, the neliösumma of a vector x = (x1, ..., xn) equals the squared length of the vector, also called the squared Euclidean norm, i.e., ||x||^2.

Classic formulae include the sum of squares of the first n natural numbers: Σ_{k=1}^n k^2 = n(n+1)(2n+1)/6. In number theory, representations of integers as sums of squares are studied; for example, Lagrange's four-square theorem states that every natural number can be expressed as a sum of four squares: a^2 + b^2 + c^2 + d^2.

In statistics, the term sum of squares is used to describe measures of variability, such as the total sum of squares in analysis of variance (ANOVA).

Etymology and usage: The compound form neliösumma is common in Finnish mathematical literature; there are also phrases like neliöiden summa. See also sum of squares, Euclidean norm, and Pythagorean theorem.