Triangular Number Calculator
Calculate the nth triangular number, the sum of the first n integers, and check whether a given number is triangular.
Last reviewed by the Radiatus Cloud team
Calculate the nth triangular number or check if a number is triangular.
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Calculate triangular numbers
A triangular number is the sum of the first n positive integers, so named because that many dots can be arranged in an equilateral triangle. This calculator finds the nth triangular number using the formula n times n plus one, divided by two, and can also test whether a given number is triangular. The sequence runs one, three, six, ten, fifteen, and so on. The tenth triangular number is fifty-five.
To check whether a number is triangular, the tool solves the formula in reverse and sees whether the index comes out as a whole number.
Triangles of numbers
Triangular numbers are among the oldest studied figurate numbers, and they appear in many places, from the handshake problem, where n people make a triangular number of handshakes, to the diagonals of Pascal triangle. The young Gauss famously summed the first hundred integers instantly by pairing them, which is exactly the triangular-number formula. They also connect to square numbers, since consecutive triangular numbers add to a perfect square.
The reverse check uses the quadratic formula, giving an exact test for whether any number belongs to the sequence. All calculation happens locally in your browser.
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Frequently Asked Questions
What is a triangular number?
The sum of the first n positive integers, which can be arranged as dots in a triangle, such as one, three, six and ten.
What is the formula?
The nth triangular number equals n times n plus one, all divided by two, a shortcut famously attributed to the young Gauss.
How do I check if a number is triangular?
Solve the formula in reverse: a number is triangular if eight times it plus one is a perfect square, which the tool tests.
Where do triangular numbers appear?
In the handshake problem, Pascal triangle, and sums of sequences, and consecutive triangular numbers add up to perfect squares.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Enter n to get the nth triangular number, or a value to test.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.