Triangular Number is a sequence of numbers that can be represented in the form of an equilateral triangle when arranged in a series. The triangular numbers list includes numbers 1, 3, 6, 10, 15… They are a type of figurative numbers. They are well known for their application in solving handshake problems.

In this article, we will learn what are triangular numbers, their definitions, examples, properties and formulas. We will also learn how to find Triangular numbers and some of their interesting facts.

What are Triangular Numbers?

Triangular numbers are numbers that can be arranged in the form of an equilateral triangle. They are a subset of figurative numbers, which are nothing but numbers that can be represented in the form of a regular shape such as a square, triangle, etc.

First triangular number is T₁ = 1.

To obtain the second number, add 2 to T_1. Thus the second number becomes 3. Subsequently, to obtain the third number, we add 3 to T₂ to arrive at number 6. For the ease of understanding, it can be represented as below:

Triangular Number Definition

Triangular number is a number represented by as many dots arranged in rows that form a triangle such that number of dots on each side are equal.

Triangular Number Examples

First five triangular numbers are:

• T₁ = 1
• T₂ = 1+2 = 3
• T₃ = 1+2+3 = 6
• T₄ = 1+2+3+4 = 10
• T₅ = 1+2+3+4+5 = 15

Visual Representation of Triangular Numbers

Graphically, triangular numbers can be represented as equilateral triangles by the number of dots equal to its numerical value. Consider the following figure:

Representation of Triangular Numbers

Triangular Number List

Triangular number list has the following numbers:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120,136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431…and so on.

Triangular Number Formula

The following formula can be used to calculate the triangular numbers:

T_n = Σⁿ_k=1 k = 1 + 2 + 3 + 4…n = n(n+1)/2

In the above formula, (n+1)/2 is binomial coefficient.

We know that sum of first ‘n’ natural numbers is given by n(n+1)/2. Hence, the sum of n natural numbers results in Triangular number.

For example, to determine the 4th triangular number, n = 4, so

T₄ = 4(4+1)/2 = 10

Hence, fourth triangular number is 10.

Triangular Number Sum

If we closely look at the pattern formed in Visual Representation of Triangular Numbers, we can easily make the following observations:

• First number is 1.
• Subsequent number is 1+2 = 3. A row is added below in the triangle with Two dots.
• Next number is 3 + 3 = 6. A row with Three dots is appended to the triangle from the step above.
• Subsequently, a new row is obtained by adding the next number in the series.

Thus the sequence can be expressed as below mentioned pattern:

1, (1+2), (1+2+3), (1+2+3+4), …

Properties of Triangular Numbers

We have learnt that triangular numbers are related to figurative numbers. They have got some interesting properties. The properties of triangular includes are discussed below in detail:

Patterns in Triangular Numbers

Triangular Numbers have a property that the number of dots equal to the numerical value of the triangular number always forms an equilateral triangle.

Relationship with Figurative Numbers

Triangular Numbers are a subset of other figurate numbers such as square, pentagon or hexagonal numbers. They have a wide variety of relations with other figurate numbers as well. Some of them are listed below:

• Sum of any two consecutive triangular number is a square number. For example, 6+10=16, which is a square of 4.
• The positive difference between any two triangular number is a trapezoidal number. For instance, 10-3=7, which is a trapezoidal number.
• Alternating triangular numbers (1, 6, 15, 28, …) are also hexagonal numbers.

Mathematical Properties of Triangular Numbers

The mathematical properties of triangular numbers are mentioned below:

• There are many triangular numbers that are also square numbers. For example 36, 1225 etc.
• Square triangular numbers can be generated from the simple recursive formula given as S_n+1 = 4S_n(8S_n + 1) where, S₁ = 1
• Square of n^th triangular number is equal to sum of the cubes of integers 1 to n. This can be expressed as Σⁿ_k=1 k³ =(Σⁿ_k=1k)²
• A positive integer n is a triangular number if and only if 8n + 1 is a square.
• Sum of first n triangular number is given by the formula {n(n + 1)(n + 2)}/6

Fibonacci Series and Triangular Numbers

Unlike triangular numbers, Fibonacci series is obtained by adding last two numbers in the sequence. For instance, 1, 1, 2, 3, 5, 8 and so on.

The only triangular number in Fibonacci Series are 1, 3, 21 and 55.

Pascal’s Triangle and Triangular Numbers

Pascal’s triangle is a triangular arrangement of numbers which are obtained in similar fashion as triangular numbers. Firstly, 1 is placed at the top. The numbers we get in subsequent steps is the addition of above two numbers. Pascal’s triangle contains two rows of all the triangular numbers, as highlighted in Fig 2.4.

How to Calculate Triangular Numbers

If we are given a sequence of triangular numbers, to determine the next number in the series, follow the below given steps:

• Calculate the difference between the two consecutive numbers.
• Increase the difference obtained by 1.
• Determine the next number in the sequence.

Consider the following series of triangular numbers.

3, 6, 10…

To determine the next number in the series:

• Step 1: Calculate the difference between two consecutive numbers. In this case it is 10-6 = 4.
• Step 2: Increment the difference obtained by 1. In our example the difference obtained is 4. thus it becomes 5 after increment.
• Step 3: Next number in the series is thus 10 + 5 = 15.

Interesting Facts About Triangular Numbers

Following are some interesting facts about triangular numbers.

• Product of two consecutive triangular numbers is never a square.
• They can solve the Handshake Problem easily. In handshake problem, we are required to find out the number of handshakes each individuals does if there are n individuals in a room.
• Digital root of all even square triangular numbers is always 9.
• Digital root of all odd square triangular numbers is always 1.
• Final digit of a triangular number is 0, 1, 3, 5, 6, or 8.

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