Tetrahedral Number — Definition, Formula & Examples

A tetrahedral number is a figurate number that represents the total count of objects stacked in the shape of a tetrahedron (a triangular pyramid). Each layer of the tetrahedron is a triangular number, and the tetrahedral number is the sum of the first

n

triangular numbers.

The $n$ -th tetrahedral number $T_n$ is defined as the sum $\sum_{k=1}^{n} \frac{k(k+1)}{2}$ , which evaluates to $\frac{n(n+1)(n+2)}{6}$ . Equivalently, $T_n = \binom{n+2}{3}$ , the binomial coefficient " $n+2$ choose 3."

Key Formula

T_n = \frac{n(n+1)(n+2)}{6}

Where:

$T_n$ = The n-th tetrahedral number
$n$ = A positive integer indicating the number of layers

How It Works

To find the

n

-th tetrahedral number, imagine stacking triangular layers. The top layer has 1 sphere, the next has 3 (a triangle of side 2), the next has 6 (side 3), and so on. Each layer

k

contains

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

objects — the

k

-th triangular number. Add these up from

k = 1

k = n

, or use the closed-form formula directly.

Worked Example

Problem: Find the 4th tetrahedral number.

Step 1: List the first 4 triangular numbers (the layers of the tetrahedron).

1, \; 3, \; 6, \; 10

Step 2: Sum them, or apply the formula directly.

T_4 = \frac{4 \cdot 5 \cdot 6}{6} = \frac{120}{6} = 20

Step 3: Verify by adding the layers:

1 + 3 + 6 + 10 = 20

Answer: The 4th tetrahedral number is 20.

Why It Matters

Tetrahedral numbers appear in combinatorics because

T_n = \binom{n+2}{3}

counts the number of ways to choose 3 items from

n+2

objects. They also show up in number theory problems involving sums of consecutive triangular numbers and in analyzing integer partitions.

Common Mistakes

Mistake: Confusing triangular and tetrahedral numbers. Students sometimes use

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

when the problem asks for a tetrahedral number.

Correction: Remember that a tetrahedral number is a sum of triangular numbers. Its formula has three consecutive factors in the numerator:

\frac{n(n+1)(n+2)}{6}

, not two.