Triangular Number Calculator
Find the nth triangular number.
Results update as you type.
How it works
Triangular numbers are a classic sequence in mathematics, and this calculator finds any term. You enter the position n, and it returns the nth triangular number and the next one in the sequence. The nth triangular number is the sum of all the natural numbers from 1 up to n, and there is a beautiful closed-form formula, n times n plus one, divided by two, that gives it directly without adding them all up. The name comes from the fact that these numbers count objects that can be arranged into an equilateral triangle: 1, 3, 6, 10, 15, and so on, like the ten pins in bowling or the fifteen balls racked in billiards. Legend has it the young mathematician Gauss discovered the formula by quickly summing 1 to 100 as 5,050, which is the 100th triangular number. Triangular numbers appear throughout mathematics: they are the entries in a diagonal of Pascal's triangle, they relate to combinations (the nth triangular number is the number of ways to choose 2 from n plus 1 items), and they connect to square and other figurate numbers. This calculator is handy for math students exploring sequences, for quickly summing consecutive integers, and for recreational math. Since the formula is exact and instant, it works for very large n. Enter a term position to find its triangular number.
T(n) = n(n + 1) / 2
The nth triangular number is the sum of the first n natural numbers, which equals n times n plus one, divided by two. It counts the objects that can be arranged in an equilateral triangle with n objects on each side, such as bowling pins or billiard balls.
Source:Number theory
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Worked example
How to use this calculator
- 1
Enter your values
Enter term (n).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
nth triangular number updates instantly as you type — no button to press.
Frequently asked questions
What is a triangular number?
A triangular number is the sum of the consecutive natural numbers from 1 up to n. The first few are 1, 3, 6, 10, and 15. They are called triangular because that many objects can be arranged into an equilateral triangle, like bowling pins (10) or a rack of billiard balls (15).
How do you calculate the nth triangular number?
Use the formula n times n plus one, divided by two. For example, the 10th triangular number is 10 times 11 divided by 2, which is 55. This closed form gives the answer instantly without adding all the numbers from 1 to n, and it works for any value of n.
What is the story about Gauss and triangular numbers?
A famous anecdote says the young mathematician Carl Friedrich Gauss, asked to add the numbers 1 to 100, found the answer 5,050 almost instantly by pairing numbers from opposite ends. That sum is exactly the 100th triangular number, illustrating the formula n times n plus one over two that this calculator uses.
Where do triangular numbers appear in math?
Triangular numbers form a diagonal of Pascal's triangle, equal the number of ways to choose 2 items from n plus 1, count handshakes among a group of people, and relate to square and other figurate numbers. They are a fundamental sequence connecting arithmetic, combinatorics, and number theory in elegant ways.
How are triangular and square numbers related?
The sum of two consecutive triangular numbers is always a perfect square. For example, the 3rd triangular number (6) plus the 4th (10) equals 16, which is 4 squared. This and other identities show how figurate numbers, numbers representing geometric arrangements of dots, interconnect across mathematics.