The smallest triangular number larger than 12 is 15.
Understanding Triangular Numbers
Triangular numbers represent the total number of items arranged in an equilateral triangle. Each triangular number is the sum of all positive integers up to a certain point. For example, the 3rd triangular number is 1 + 2 + 3 = 6.
The Formula for Triangular Numbers
The nth triangular number, denoted as T_n, can be calculated using the formula:
$$T_n = \frac{n(n+1)}{2}$$
Where 'n' is a positive integer representing the position of the triangular number in the sequence.
Finding the Smallest Triangular Number Greater Than 12
To find the smallest triangular number that exceeds 12, we can list the first few triangular numbers in sequence:
- T_1: 1 = 1(1+1)/2
- T_2: 1 + 2 = 3 = 2(2+1)/2
- T_3: 1 + 2 + 3 = 6 = 3(3+1)/2
- T_4: 1 + 2 + 3 + 4 = 10 = 4(4+1)/2
- T_5: 1 + 2 + 3 + 4 + 5 = 15 = 5(5+1)/2
We can also visualize this progression in a table:
| n | Triangular Number (T_n = n(n+1)/2) |
|---|---|
| 1 | 1 |
| 2 | 3 |
| 3 | 6 |
| 4 | 10 |
| 5 | 15 |
| 6 | 21 |
As shown in the table and calculations, the triangular number for n=4 is 10, which is not larger than 12. The next triangular number, for n=5, is 15. This is the first triangular number in the sequence that is greater than 12.
Therefore, the smallest triangular number exceeding 12 is 15.
