What this calculator does
A triangular number counts how many dots it takes to arrange them in a triangle, one dot in the first row, two in the second, three in the third, and so on. The 6th triangular number, for example, is 21: the total from stacking rows of 1, 2, 3, 4, 5 and 6 dots on top of each other.
The formula, n(n+1)/2, is the same one used to work out the sum of the first n whole numbers, because that is exactly what a triangular number is. It is worth knowing partly because it turns up again inside other formulas, such as counting the handshakes possible in a group or the diagonals of a shape, wherever something grows by adding one more each step.
The formula
The nth triangular number is n multiplied by (n+1), divided by 2. This comes from pairing the sequence 1 to n with itself in reverse: adding 1+n, 2+(n-1), 3+(n-2) and so on always gives n+1, and there are n/2 such pairs, so the total is n(n+1)/2.
| Term | Meaning |
|---|---|
| Tₙ | The nth triangular number, the sum of every whole number from 1 to n. |
| n | The term number: which triangular number in the sequence is being found. |
| Tₙ₋₁ | The previous triangular number in the sequence, one row smaller. |
The inputs explained
| Field | What to enter |
|---|---|
| Term number (n) | Enter the term number as a positive whole number: 6 finds the 6th triangular number. |
When to use it
Working out how many games a round robin needs
If every team in a group plays every other team once, the number of matches is a triangular number one less than the team count, since it is the same sum-of-1-to-n pattern applied to how many opponents remain unpaired at each step.
Stacking objects in a triangular display
Anyone arranging cans, balls or cups into a triangular pyramid layer by layer is building up triangular numbers one row at a time, and this calculator gives the running total at any row.
Checking a sequence in a maths problem
Triangular numbers appear often in number-pattern questions. Confirming a term against the formula catches an arithmetic slip before it carries through the rest of a problem.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What are the first several triangular numbers?
The nth triangular number and the term immediately before it, across a spread of values for n.
Questions
What is the first triangular number?
T₁ is 1, a single dot, since a triangle of one row has nothing to add to it. The sequence then continues 1, 3, 6, 10, 15, 21 and so on as each new row is added.
Can n be zero or negative?
No. The triangular number sequence starts at n = 1, describing a triangle with at least one row, so this calculator requires a whole number of 1 or more.
How is this different from a square number?
A square number arranges dots into an n-by-n grid (n²); a triangular number arranges them into a triangle with rows of increasing length (n(n+1)/2). The two sequences grow at different rates and only share the value 1 at the very start.
Is there a quick way to check if a number is triangular?
A number x is triangular exactly when 8x+1 is a perfect square, which comes from solving n(n+1)/2 = x for n using the quadratic formula. The perfect square calculator can confirm the 8x+1 part of that check.
To check whether the perfect-square test above actually holds for a given number, use the perfect square calculator. For a general sequence of numbers each growing by a fixed amount rather than an increasing amount, see the arithmetic sequence calculator.