Revision as of 07:15, 19 May 2008 by imported>Karsten Meyer
A triangular number represents the number of circles you can arrange to a equilateral triangle.
Definition
Properties
The triangular number is related to many other figurated numbers:
- The sum of two consecutive triangles is a square number:
![{\displaystyle \Delta _{n-1}+\Delta _{n}={\frac {(n-1)\cdot n}{2}}+{\frac {n\cdot (n+1)}{2}}={\frac {n\cdot (n-1+n+1)}{2}}={\frac {n\cdot (2n)}{2}}={\frac {2n^{2}}{2}}=n^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4606885bdfa136f2138c198aa70b6e6ad28e8318)
is a centered square number
is a centered hexagonal number
is an odd square number
![{\displaystyle \sum _{i=1}^{n}i^{3}=\Delta _{n}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86ea61778a64746a6ed57fbcd7a4142e9067ec55)
Every even perfect number is a triangular number