Triangular number
A triangular number, Tn is a type of figurate number (a number that can be represented using a regular geometric pattern formed using dots that are regularly spaced). Triangular numbers are numbers that, when represented using regularly spaced dots, form an equilateral triangle. Other examples of figurate numbers include square numbers and pentagonal numbers.
The first 30 triangular numbers, starting from T1, are listed below:
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.
Triangular numbers can be determined as the sum of natural numbers from 1 to n:
T1 = 1
T2 = 1 + 2
T3 = 1 + 2 + 3
This is reflected in the rows of the triangle. For T1 there is only one row. For T2, we add a second row (2 dots) to T1 (1 dot), to find that the 2nd triangular number, T2 = 3. Each subsequent triangular number adds 1 row to the previous triangular number, where the new row has 1 additional dot, and the total number of dots in the new row is equal to n, as shown in the figure below.
The row of red dots represents the new row in each respective triangular number, where the number of dots is equal to n. There are infinitely many triangular numbers, so using the above method to find triangular numbers is exceedingly cumbersome. Fortunately, there is a formula to determine Tn:
Examples
Find Tn for n = 1, 3, and 5 using the formula.
1. 
2. 
3. 
How to determine if a number is triangular
It is possible to test whether or not a given integer, x, is a triangular number using some basic algebra and square numbers. An integer, x, is a triangular number, if and only if the result of 8x + 1 is a square number.
Examples
Test whether the following numbers are triangular numbers.
1. x = 6:
49 is a square number since it is the product of 7 × 7, so 6 is a triangular number.
2. x = 21:
169 is a square number since it is the product of 13 × 13, so 21 is a triangular number.
3. x = 45:
361 is a square number since it is the product of 19 × 19, so 45 is a triangular number.
Triangular numbers and other figurate numbers
Triangular numbers have a variety of relationships to other figurate numbers, such as square numbers, pentagonal numbers, and hexagonal numbers.
Square numbers
The sum of two consecutive triangular numbers is a square number. Algebraically, this is written as:
Example
For n = 2:
A square number is found as n2, so in this case 22 = 4, confirming the relationship.
Pentagonal numbers
Triangular numbers are related to pentagonal numbers, Pn by the following formula:
formula:
Hexagonal numbers
Triangular numbers are related to hexagonal numbers, Hn, by the following formula:
