Triangular Numbers Calculator

What are triangular numbers?

A triangular number or triangle number counts the objects that can form an equilateral triangle. The nth triangle number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n.

The general representation of a triangular number is

T_n= 1 + 2 + 3 + 4 +...+ (n-2) + (n-1) + n,

where n is a natural number.

This sum is T_n = n * (n + 1) / 2. This is the triangular number formula to find the nth triagular number.

To prove that this formula is true, write twice the general representation and rearange the terms as below

T_n = 1 + 2 + 3 + ...+ (n-2) + (n-1) + n
T_n = n + (n-1) + (n-2) +... + 3 + 2 + 1
---------------------------------------
T_n = (1 + n) + 2 + ( n - 1) + 3 + (n - 2) + ...+ (n-1) + 2 + n + 1
T_n = (1 + n) + (1 + n) + (1 + n) +...+ (1 + n) + (1 + n). There are n terms, so

2T_n = n * (n+1) or T_n = n * (n + 1) / 2