How to Find Triangular Numbers

A triangular number (also known as triangle number) include objects organized in an equilateral triangle. The nth triangular number is the number of black dots in the triangular pattern with n black dots on a side and is equivalent to the total of the “n” natural numbers from “1” to “n”. The arrangement of triangular numbers, beginning at the 0th triangular number, is:

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630,

The triangle numbers can be calculated by the formulas given below:

T n = \sum k = 1 n k = 1 + 2 + 3 + . . . + n = n ( n + 1 ) 2 = (n + 1 2) Tn=\sumk=1nk=1+2+3+\dots+n=n(n+1)2=(n+12)

Where (n+1)/2 is the Binomial coefficient. It shows the number of distinct pairs that can be chosen from (n + 1) objects, and it is said as “n plus one to one choose two”.

The first equation can be represented using an image. For every triangular number Tn, think of a “half-square” pattern of objects corresponding to the triangular number, as shown in the image below. Replicating this pattern and turning it upside down to create a rectangular image doubles the number of objects, giving a rectangle with dimensions n×(n+1), which is also the number of objects in the rectangle. It clearly shows that the triangular number itself is in every case precisely half of the number of objects in such an image or Tn= (n (n+1)/2).