Triangular Numbers Calculator

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. Also try the calculate Integer.

Enter a positive integer to find the triangular number at that position.

Enter any whole number to test whether it belongs to the triangular sequence.

How many triangular numbers to list in the sequence table.

Results

Triangular Number T(n)

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Is It Triangular?

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Its Position in Sequence

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Sum Expression

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Results Table

Frequently Asked Questions

What are triangular numbers?

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. See also our use the GCF and LCM Calculator.

How do I find the nth triangular number?

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.

What are the first triangular numbers?

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.

How do I check if a number is triangular?

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. You might also find our Decimal Calculator useful.

Why is 1 a triangular number?

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.

Do triangular numbers have real-world applications?

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.

What is the relationship between triangular numbers and square numbers?

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.

Are there infinitely many triangular 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.