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Calculate Triangular Number

Calculator for the triangular number at a given place, using Gauss's summation formula. A triangular number is the sum of consecutive natural numbers, starting at 1 (or at 0). The second triangular number is 1+2=3, the third is 1+2+3=6, the fourth triangular number is 1+2+3+4=10 and so on. The name is triangular number, because if you want to build an equilateral triangle with equal pieces, the number of pieces needed is a triangular number.
Please enter a natural number. The triangular number on this place will be calculated.

The formula for the n-th triangular number is n*(n+1)/2.

Place:

The n. triangular number is .

Example: the 27. triangular number is 378.

A few interesting details about triangular numbers:
• Triangular numbers are often represented by a triangle symbol △, △₁ for the first triangular number, △₂ for the second, and so on.
• The inverse of the triangular number function is the triangular root. If you substitute a triangular number into the function [√(8△+1) - 1]/2, you obtain the position of that triangular number.
• Every natural number can be expressed as the sum of three or less triangular numbers. This was conjectured by Pierre de Fermat and proven by Carl Friedrich Gauss in the 18th century.
• The infinite sum of the reciprocals of all triangular numbers is two, Σ 1/△ = 2, according to Gottfried Wilhelm Leibniz.
• The tenth triangular number is 55, the hundredth is 5,050, the thousandth is 500,500, and so on.
• Analogous to triangular numbers, there are also polygonal numbers, square numbers, pentagonal numbers, hexagonal numbers, and so forth. Of these, square numbers 1, 4, 9, 16, ... are of course the best known and by far the most widely used. Another generalization arises from adding further dimensions. In three dimensions, for instance, there are tetrahedral numbers. This principle can be extended to all five Platonic solids. All these numbers can be calculated using binomial coefficients.

Last updated on 08/26/2026. Author: Jürgen Kummer