Calculate the nth triangular number instantly using the formula \(T(n)=n(n+1)/2\). See the calculation, triangular number sequence, examples, and learn how to identify triangular numbers.
Enter a positive integer \(n\) to calculate \(T(n)\).
The calculator accepts integers from 1 to 1,000,000,000. The complete sequence is displayed only for smaller inputs.
A triangular number is a number that can be represented as the sum of the first \(n\) positive integers.
The first triangular numbers are:
For example:
Therefore, 15 is the fifth triangular number: \[ T(5)=15 \]
The standard formula for the nth triangular number is:
This formula is equivalent to the sum:
The formula allows you to calculate a triangular number directly without adding every integer from 1 through \(n\).
| n | T(n) |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 6 |
| 4 | 10 |
| 5 | 15 |
| 6 | 21 |
| 7 | 28 |
| 8 | 36 |
| 9 | 45 |
| 10 | 55 |
| 11 | 66 |
| 12 | 78 |
| 13 | 91 |
| 14 | 105 |
| 15 | 120 |
| 16 | 136 |
| 17 | 153 |
| 18 | 171 |
| 19 | 190 |
| 20 | 210 |
| 21 | 231 |
| 22 | 253 |
| 23 | 276 |
| 24 | 300 |
| 25 | 325 |
A positive integer \(x\) is a triangular number if and only if \(8x+1\) is a perfect square.
This follows from solving:
For example, consider \(15\):
Therefore 15 is triangular, and its index is:
A triangular number is a number that can be written as the sum of the first \(n\) positive integers. The nth triangular number is \(T(n)=n(n+1)/2\).
The formula is: \[ T(n)=\frac{n(n+1)}{2}. \]
The 10th triangular number is: \[ T(10)=\frac{10(11)}{2}=55. \]
Yes. 15 is the fifth triangular number because \(1+2+3+4+5=15\).
No. Although 100 is a positive integer, it does not satisfy the triangular-number test because \(8(100)+1=801\), which is not a perfect square.
Calculate \(8x+1\). If the result is a perfect square, then \(x\) is a triangular number.
A triangular number is one particular value such as 15. The triangular number sequence is the complete sequence \(1,3,6,10,15,21,\ldots\).