A triangular number is a number that can be represented by a pattern of dots arranged in an equilateral triangle with the same number of dots on each side.
For example:
The first triangular number is 1, the second is 3, the third is 6, the fourth 10, the fifth 15, and so on.
You can see that each triangle comes from the one before by adding a row of dots on the bottom which has one more dot than the previous bottom row. This means that the 
triangular number is equal to
There’s also another way we can calculate the triangular number. Take two copies of the dot pattern representing the triangular number and arrange them so that they form a rectangular dot pattern.
This rectangular pattern will have dots on the shorter side and dots on the longer side, which means that the rectangular pattern contains dots in total. And since the original triangular dot pattern constitutes exactly half of the rectangular pattern, we know that the triangular number is
Note that with this consideration we have proved the formula for the summation of natural numbers, namely
Triangular numbers have lots of interesting properties. For example, the sum of consecutive triangular numbers is a square number. You can see this by arranging the triangular dot patterns representing the and triangular numbers to form a square which has dots to a side:
Alternatively, you can see this using the formulas for the consecutive triangular numbers and :
What is more, alternating triangular numbers (1, 6, 15, ...) are also hexagonal numbers (numbers formed from a hexagonal dot pattern) and every even perfect number is a triangular number.
Triangular numbers also come up in real life. For example, a network of computers in which every computer is connected to every other computer requires connections. And if in sports you are playing a round robin tournament, in which each team plays each other team exactly once, then the number of matches you need for teams is These two results are equivalent to the handshake problem we have explored on Plus before.
We would like to thank Zoheir Barka who sent us the first draft of this article. We will publish a lovely article by Barka about triangular numbers soon. In the mean time, you can read Barka's article about beautiful patterns in multiplication tables here.
FAQs
The first triangular number is 1, the second is 3, the third is 6, the fourth 10, the fifth 15, and so on. triangular number and arrange them so that they form a rectangular dot pattern.
What is the quick way to work out triangular numbers? ›
The formula for expressing how to find a triangular number is known as n(n + 1)/ 2. For example, if we're looking to find the fifth triangular number, we replace n with the number 5. This turns the formula into 5 (5 + 1) / 2.
What is the rule for triangular numbers? ›
Triangular numbers are numbers that make up the sequence 1, 3, 6, 10, . . .. The nth triangular number in the sequence is the number of dots it would take to make an equilateral triangle with n dots on each side. The formula for the nth triangular number is (n)(n + 1) / 2.
What is the triangular pattern in math? ›
The triangular number sequence is the representation of the numbers in the form of equilateral triangle arranged in a series or sequence. These numbers are in a sequence of 1, 3, 6, 10, 15, 21, 28, 36, 45, and so on. The numbers in the triangular pattern are represented by dots.
What are the triangular numbers 1 to 50? ›
0, 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...
What is the algorithm for triangular numbers? ›
The trick can be expressed as a simple rule (the algorithm): For the nth triangular number just multiply n by (n+1) then divide by 2.
What is the formula for a triangular? ›
The area of a triangle is defined as the total region that is enclosed by the three sides of any particular triangle. Basically, it is equal to half of the base times height, i.e. A = 1/2 × b × h.
What is the formula for triangle counting? ›
The number of triangles can be obtained easily by using a simple formula which is total number of triangles = [n(n+2)(2n+1)] / 8 where 'n' is the number of triangles in sides. In the given figure, we can see that the number of triangles on each side is 4.
What are some interesting facts about triangular numbers? ›
There are infinitely many triangular numbers such as t8 = 36 that are squares. The sum of the reciprocals of the triangular numbers is 2. Theorem:The n-th square and the n-th oblong number add up to a triangular number.
What is considered the perfect number? ›
perfect number, a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128.
We have T(n+1)=T(n)+t(n+1). nth triangular number is the sum of n consecutive natural numbers from starting which is simply n(n+1)/2. You want sum of first n triangular numbers.
What is a happy number in maths? ›
In number theory, a happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because , and . On the other hand, 4 is not a happy number because the sequence starting with and eventually reaches.
What is the formula for the number of triangles? ›
The number of triangles can be obtained easily by using a simple formula which is total number of triangles = [n(n+2)(2n+1)] / 8 where 'n' is the number of triangles in sides. In the given figure, we can see that the number of triangles on each side is 4. Therefore, total number of triangles = 27.
What is the efficient way to find triangular square numbers? ›
First few Squared triangular number are 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, … A simple solution is to one by one add cubes of natural numbers. If current sum becomes same as given number, then we return count of natural numbers added so far. Else we return -1.
What is the formula for finding a triangular? ›
Basically, it is equal to half of the base times height, i.e. A = 1/2 × b × h. Hence, to find the area of a tri-sided polygon, we have to know the base (b) and height (h) of it. It is applicable to all types of triangles, whether it is scalene, isosceles or equilateral.
triangular number 
is equal to 
dots on the shorter side and 
dots on the longer side, which means that the rectangular pattern contains 
dots in total. And since the original triangular dot pattern constitutes exactly half of the rectangular pattern, we know that the 
triangular number 
is 
natural numbers, namely 
and 
triangular numbers to form a square which has 
dots to a side:
and 
: 
computers in which every computer is connected to every other computer requires 
connections. And if in sports you are playing a round robin tournament, in which each team plays each other team exactly once, then the number of matches you need for 
teams is 
These two results are equivalent to the handshake problem we have explored on Plus before. 