What Is a Triangular Number?

A triangular number is a number that can be arranged in the shape of a triangle. It is formed by adding natural numbers in order. The following example shows how triangular numbers form a triangle shape:

   *         → 1st triangular number (1)
  * *        → 2nd triangular number (1 + 2 = 3)
 * * *       → 3rd triangular number (1 + 2 + 3 = 6)

Definition

A triangular number is defined as the sum of the first n natural numbers.
It can be calculated using the formula:

T_n = 1 + 2 + 3 + \cdots + n = \frac{n(n + 1)}{2}

Examples

Order $n$	Triangular Number $T_n$	Formula
1	1	$\frac{1×2}{2} = 1$
2	3	$\frac{2×3}{2} = 3$
3	6	$\frac{3×4}{2} = 6$
4	10	$\frac{4×5}{2} = 10$
5	15	$\frac{5×6}{2} = 15$