Triangular & Tetrahedral Number Calculator

Calculate the nth triangular number and nth tetrahedral number instantly. Learn the formulas, see examples and sequences, and understand how triangular and tetrahedral numbers are related.

Calculate a Triangular and Tetrahedral Number

Enter a positive integer \(n\). The calculator finds both \[ T(n)=\frac{n(n+1)}{2} \] and \[ Te(n)=\frac{n(n+1)(n+2)}{6}. \]

What Is a Triangular Number?

A triangular number is a number that can be represented as a triangular arrangement of objects. Algebraically, it is the sum of the first \(n\) positive integers.

\[ T(n) = 1+2+3+\cdots+n = \frac{n(n+1)}{2} \]

The first triangular numbers are

\[ 1,\;3,\;6,\;10,\;15,\;21,\;28,\;36,\;45,\;55,\ldots \]

For example, \(T(5)=1+2+3+4+5=15\).

What Is a Tetrahedral Number?

A tetrahedral number is the sum of the first \(n\) triangular numbers. It corresponds to arranging objects in a three-dimensional tetrahedral pattern.

\[ Te(n) = T(1)+T(2)+\cdots+T(n) \]

Because \[ T(i)=\frac{i(i+1)}{2}, \] the tetrahedral number has the closed-form formula

\[ Te(n) = \frac{n(n+1)(n+2)}{6}. \]

The first tetrahedral numbers are

\[ 1,\;4,\;10,\;20,\;35,\;56,\;84,\;120,\;165,\;220,\ldots \]

Relationship Between Triangular and Tetrahedral Numbers

The two sequences are directly connected. A tetrahedral number is the cumulative sum of triangular numbers:

\[ Te(n) = \sum_{i=1}^{n}T(i) \]

For example:

\[ T(1)+T(2)+T(3)+T(4) = 1+3+6+10 = 20 \]

Therefore, \(Te(4)=20\).

Examples

Example 1: \(n=5\)

The fifth triangular number is

\[ T(5) = \frac{5(6)}{2} = 15 \]

The fifth tetrahedral number is

\[ Te(5) = \frac{5(6)(7)}{6} = 35 \]

Example 2: \(n=10\)

\[ T(10) = \frac{10(11)}{2} = 55 \]

\[ Te(10) = \frac{10(11)(12)}{6} = 220 \]

Where Are These Numbers Used?

Triangular and tetrahedral numbers are examples of figurate numbers. They appear in several areas of mathematics.

Number theory: studying special integer sequences and their properties.
Combinatorics: counting pairs, triples and other structured collections.
Geometry: representing objects arranged in triangular or tetrahedral patterns.
Discrete mathematics: describing cumulative sums and structured counting problems.

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Frequently Asked Questions

What is a triangular number?

A triangular number is the sum of the first n positive integers. The nth triangular number is T(n) = n(n + 1)/2.

What is a tetrahedral number?

A tetrahedral number is the sum of the first n triangular numbers. The nth tetrahedral number is Te(n) = n(n + 1)(n + 2)/6.

What is the formula for the nth triangular number?

The formula is T(n) = n(n + 1)/2.

What is the formula for the nth tetrahedral number?

The formula is Te(n) = n(n + 1)(n + 2)/6.

How are triangular and tetrahedral numbers related?

The nth tetrahedral number is the sum of the first n triangular numbers: Te(n) = T(1) + T(2) + ... + T(n).

What are the first triangular numbers?

The first triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, and 55.

What are the first tetrahedral numbers?

The first tetrahedral numbers are 1, 4, 10, 20, 35, 56, 84, 120, 165, and 220.