Triangular Number Calculator
Find the nth triangular number, list the sequence T(1…n), and test whether any value is triangular.
Sum of all integers from 1 to 10
- 1
n + 1
10 + 1 = 11 - 2
n × (n + 1)
10 × 11 = 110 - 3
T(n) = n × (n + 1) ÷ 2
110 ÷ 2 = 55The Gauss formula: T(n) equals the sum 1 + 2 + … + n.
How does this calculator work?
The nth triangular number is T(n) = n(n+1)/2, the sum 1+2+3+…+n. Enter n to get T(n) instantly and see the full sequence plotted. A number x is triangular when 8x+1 is a perfect square; for example 55 is triangular because 8×55+1 = 441 = 21², making it the 10th triangular number.
Formula
How this is calculated
Enter a positive integer n (the term index). The nth triangular number is the sum of the first n natural numbers, 1 + 2 + … + n. Instead of adding term by term, the calculator uses the closed-form Gauss formula T(n) = n(n+1)/2, which is exact and instantaneous for any n.
The calculator also builds the running sequence T(1), T(2), …, T(n) and plots it. Because T(n) grows like n²/2, the area curve has a clear quadratic (upward-bending) shape. For display performance the listed/plotted sequence is capped at the first 200 terms, but the headline T(n) is always computed directly from the formula for the exact n you entered.
The optional test value uses the number-theory fact that a non-negative integer x is triangular if and only if 8x + 1 is a perfect square. When it is, the term index is recovered as (√(8x+1) − 1) / 2. Inputs are treated as integers; fractional or negative n are rejected, and the test only applies to non-negative integers.
Frequently asked questions
It is the sum 1 + 2 + 3 + … + n, equal to n(n+1)/2. For example T(5) = 5×6/2 = 15.
A non-negative integer x is triangular exactly when 8x + 1 is a perfect square. For instance 8×55 + 1 = 441 = 21², so 55 is triangular (it is T(10)).
Triangular numbers grow quadratically: T(n) = n(n+1)/2 ≈ n²/2, so each step adds a larger amount and the plotted curve bends upward.
Also known as
TG we-Calculate Editorial Team. (2026). Triangular Number Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/triangular-number-calculator
TG we-Calculate Editorial Team. "Triangular Number Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/triangular-number-calculator.
TG we-Calculate Editorial Team, "Triangular Number Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/triangular-number-calculator
@misc{wecalculate_triangular_number_calculator, title = {Triangular Number Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/triangular-number-calculator}}, year = {2026}, note = {TG we-Calculate} }
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