Triangular Number — Definition, Formula & Examples

A triangular number is the total number of dots needed to form an equilateral triangle with n rows, where each row has one more dot than the row above it. The first few triangular numbers are 1, 3, 6, 10, 15, 21, 28, …

The $n$ th triangular number $T_n$ is defined as the sum of the first $n$ positive integers: $T_n = 1 + 2 + 3 + \cdots + n = \dfrac{n(n+1)}{2}$ for any positive integer $n$ .

Key Formula

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

Where:

$T_n$ = The nth triangular number
$n$ = A positive integer representing the number of rows (or the position in the sequence)

How It Works

To find the

n

th triangular number, plug

n

into the formula

\frac{n(n+1)}{2}

. You can also think of it as stacking rows: row 1 has 1 dot, row 2 has 2 dots, row 3 has 3 dots, and so on. The total dot count after

n

rows is your triangular number. A famous story attributes this shortcut to young Gauss, who noticed that pairing the first and last terms of

1 + 2 + \cdots + 100

always gives 101, and there are 50 such pairs, yielding 5050.

Worked Example

Problem: Find the 8th triangular number.

Apply the formula: Substitute n = 8 into the formula.

T_8 = \frac{8 \times (8+1)}{2}

Simplify: Multiply 8 by 9, then divide by 2.

T_8 = \frac{72}{2} = 36

Answer: The 8th triangular number is 36.

Visualization

Why It Matters

Triangular numbers appear in combinatorics:

T_n

equals

\binom{n+1}{2}

, the number of ways to choose 2 items from

n+1

objects. They also show up when counting handshakes in a group, stacking objects in pyramids, and in number theory problems involving sums and partitions.

Common Mistakes

Mistake: Using

n^2/2

instead of

n(n+1)/2

Correction: The formula requires multiplying

n

by the next integer

n+1

before dividing by 2. For example,

T_4 = 4 \times 5 / 2 = 10

, not

4^2 / 2 = 8