triangsum

Triangsum is a mathematical concept that refers to the sum of consecutive integers starting from 1 up to a given positive integer, n. This sum can be represented by the formula: 1 + 2 + 3 + ... + n. The value of this sum has a well-known closed-form expression, which is n(n+1)/2. This formula allows for the efficient calculation of the triangsum without needing to add each individual number.

The name "triangsum" is derived from the fact that these sums correspond to the number of dots required to form equilateral triangles. For example, a triangle with one dot on each side has a triangsum of 1, a triangle with two dots on each side has a triangsum of 1+2=3, and a triangle with three dots on each side has a triangsum of 1+2+3=6. These numbers are also known as triangular numbers.

Triangsums appear in various mathematical contexts, including number theory and combinatorics. They are related to the arithmetic progression and are a fundamental example of a sum of a sequence. The concept is often introduced early in mathematics education to illustrate the power of formulas and the properties of number sequences.