A triangular number is the sum of N consecutive integers starting from 1.
Since the numbers are consecutive, each number is greater than 1 compared to the previous one. The triangular number therefore owes its name to its representation in the form of a triangle, where each line corresponds to the number gradually added up and the number of lines, or the height of the triangle, corresponds to the position of the number in the succession of triangular numbers.
The first triangular numbers are:
1 : 1
3 : 1 + 2
6 : 1 + 2 + 3
10: 1 + 2 + 3 + 4
15: 1 + 2 + 3 + 4 + 5
The sum of two equal triangular numbers of position N gives a rectangle of sides N and N + 1. Its area is therefore N * (N + 1). From here we get the Gauss sum, which is used to calculate the sum of a set of consecutive numbers.
The sum of two consecutive triangular numbers gives a square number. Visually it is clear: a triangular number can be drawn upside down but smaller than a line to obtain a square.