Triangular numbers count dots arranged in triangles: 1, 3, 6, 10, 15 … They count handshakes in a group (T₍ₙ₋₁₎), bowling pins (T₄ = 10) and appear in Pascal’s triangle.
How to use the Triangular Numbers Calculator
- Number n
- Is this number triangular?
Results update as you type, and everything is calculated in your browser.
How it works
Gauss’s trick pairs the numbers 1 … n from both ends to get n(n + 1)/2. The inverse formula tells you whether a number is triangular and, if so, which one.
Inputs explained
| Field | What to enter |
|---|---|
| Number n | A number |
| Is this number triangular? | A number |
What each result means
- Triangular number Tₙ = [Number n]×([Number n] + 1)÷2
- Next triangular number Tₙ₊₁ = ([Number n] + 1)×([Number n] + 2)÷2
- Its index k (Tₖ = number) = (sqrt(8×[Is this number triangular?] + 1) – 1)÷2
Example values
How the main result changes when Number n changes and everything else stays at the example values above:
| Number n | Triangular number Tₙ |
|---|---|
| 6 | 21 |
| 9 | 45 |
| 12 | 78 |
| 15 | 120 |
| 18 | 171 |
| 24 | 300 |
Formula
Tₙ = 1 + 2 + … + n = n(n + 1) / 2
x is triangular ⇔ (√(8x + 1) − 1)/2 is a whole number
Worked example
Inputs — Number n: 12; Is this number triangular?: 78.
Results — Triangular number Tₙ = 78; Next triangular number Tₙ₊₁ = 91; Is the second number triangular? = Yes; Its index k (Tₖ = number) = 12.
Key terms
- Triangular number
- A number of dots that can form an equilateral triangle.
Frequently asked questions
What are the first triangular numbers?
1, 3, 6, 10, 15, 21, 28, 36, 45, 55.
How many handshakes happen among 10 people?
T₉ = 45.
What do two consecutive triangular numbers add to?
A perfect square: Tₙ + Tₙ₋₁ = n².
Is 78 triangular?
Yes, T₁₂ = 78.