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Concept
Triangular number
Summary
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The nth triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is
The triangular numbers are given by the following explicit formulas:
where , does not mean division, but is the notation for a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as "n plus one choose two".
The first equation can be illustrated using a visual proof. For every triangular number , imagine a "half-rectangle" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions , which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or: . The example follows:
This formula can be proven formally using mathematical induction. It is clearly true for :
Now assume that, for some natural number , . Adding to this yields
so if the formula is true for , it is true for . Since it is clearly true for , it is therefore true for , , and ultimately all natural numbers by induction.
The German mathematician and scientist, Carl Friedrich Gauss, is said to have found this relationship in his early youth, by multiplying n/2 pairs of numbers in the sum by the values of each pair n + 1. However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans in the 5th century BC. The two formulas were described by the Irish monk Dicuil in about 816 in his Computus.
Official source
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