Definition:Triangular Number/Definition 2

Definition

Triangular numbers are defined as:

$\ds T_n = \sum_{i \mathop = 1}^n i = 1 + 2 + \cdots + \paren {n - 1} + n$

for $n = 1, 2, 3, \ldots$

Examples of Triangular Numbers

The first few triangular numbers are as follows:

Sequence of Triangular Numbers

The sequence of triangular numbers, for $n \in \Z_{\ge 0}$, begins:

$0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, \ldots$

Also see

• Equivalence of Definitions of Triangular Number

Sources

• 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.3$ Early Number Theory: Problems $1.3$: $1$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $15$
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.2$: Pythagoras (ca. $\text {580}$ – $\text {500}$ B.C.)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $15$