Triangular Numbers Table
A complete reference table for triangular numbers T(1) through T(50). A triangular number counts the objects that can be arranged into an equilateral triangle — the n-th triangular number is the sum of the first n positive integers, given by the formula T(n) = n(n+1)/2.
Triangular Number Calculator
Find the nth triangular number, T(n) = n(n + 1) / 2.
T(10)55
What is a Triangular Number?
The n-th triangular number T(n) is defined as:
- T(n) = 1 + 2 + 3 + … + n = n(n+1) / 2
- T(1) = 1, T(2) = 3, T(3) = 6, T(4) = 10, T(5) = 15, …
Each value equals the previous triangular number plus n: T(n) = T(n−1) + n. This means the difference column in the table below simply counts up by 1 each row.
Triangular Numbers T(1) to T(50)
The table shows n, T(n), and the difference T(n) − T(n−1) = n to illustrate how each term is built from the previous one.
| n | T(n) | n | T(n) |
|---|---|---|---|
| 1 | 1 | 26 | 351 |
| 2 | 3 | 27 | 378 |
| 3 | 6 | 28 | 406 |
| 4 | 10 | 29 | 435 |
| 5 | 15 | 30 | 465 |
| 6 | 21 | 31 | 496 |
| 7 | 28 | 32 | 528 |
| 8 | 36 | 33 | 561 |
| 9 | 45 | 34 | 595 |
| 10 | 55 | 35 | 630 |
| 11 | 66 | 36 | 666 |
| 12 | 78 | 37 | 703 |
| 13 | 91 | 38 | 741 |
| 14 | 105 | 39 | 780 |
| 15 | 120 | 40 | 820 |
| 16 | 136 | 41 | 861 |
| 17 | 153 | 42 | 903 |
| 18 | 171 | 43 | 946 |
| 19 | 190 | 44 | 990 |
| 20 | 210 | 45 | 1,035 |
| 21 | 231 | 46 | 1,081 |
| 22 | 253 | 47 | 1,128 |
| 23 | 276 | 48 | 1,176 |
| 24 | 300 | 49 | 1,225 |
| 25 | 325 | 50 | 1,275 |
Key Properties
- Consecutive sum: T(n) + T(n−1) = n² — every perfect square is the sum of two consecutive triangular numbers.
- Combinatorial identity: T(n) = C(n+1, 2) — the number of ways to choose 2 items from n+1.
- 8T(n) + 1 is always a perfect square: e.g. 8×6 + 1 = 49 = 7².
- Square triangular numbers: numbers that are both square and triangular: 1, 36, 1,225, 41,616, …
- Gauss's formula: T(n) = n(n+1)/2 was famously derived by Carl Friedrich Gauss as a child to quickly sum 1 through 100.
- Every positive integer is the sum of at most three triangular numbers (Gauss's Eureka theorem).