Enter a position number n to find the nth triangular number, or enter any number to check if it's triangular. The Triangular Numbers Calculator returns the triangular number value, confirms membership in the sequence, and shows the first several triangular numbers in a visual list.
Triangular Number T(n)
--
Is It Triangular?
--
Its Position in Sequence
--
Sum Expression
--
Triangular numbers are the sum of consecutive natural numbers starting from 1. For example, T(4) = 1+2+3+4 = 10. They get their name because that many dots can always be arranged into an equilateral triangle shape.
Use the formula T(n) = n × (n + 1) / 2. For example, the 10th triangular number is 10 × 11 / 2 = 55. This formula works because you are summing an arithmetic sequence from 1 to n.
The first ten triangular numbers are: 1, 3, 6, 10, 15, 21, 28, 36, 45, and 55. Each one is formed by adding the next natural number to the previous triangular number.
A number x is triangular if 8x + 1 is a perfect square. If √(8x + 1) is a whole number, then x is triangular and its position n = (√(8x+1) − 1) / 2. This calculator performs that check automatically.
1 is the first triangular number because T(1) = 1 × (1+1) / 2 = 1. A single dot trivially forms a triangle, and the sequence of cumulative sums begins at 1.
Yes — triangular numbers appear in combinatorics (handshake problems), computer science (triangular arrays), physics (energy levels), and even bowling (10 pins form T(4) = 10). They're foundational in number theory and combinatorial mathematics.
The sum of any two consecutive triangular numbers is always a perfect square. For example, T(3) + T(4) = 6 + 10 = 16 = 4². This elegant property connects triangular and square figurate numbers.
Yes — for every positive integer n there is a corresponding triangular number T(n) = n(n+1)/2, so the sequence is infinite. The numbers grow roughly as n²/2, getting increasingly spread apart as n increases.