User:Acwir/แซนด์บ็อกซ์

The pattern found for triangular numbers $\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{2}+1}{2}}$ and for tetrahedral numbers $\sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{3}+2}{3}},$ which uses binomial coefficients, can be generalized. This leads to the formula:

$$\sum _{n_{k-1}=1}^{n_{k}}\sum _{n_{k-2}=1}^{n_{k-1}}\ldots \sum _{n_{2}=1}^{n_{3}}\sum _{n_{1}=1}^{n_{2}}n_{1}={\binom {n_{k}+k-1}{k}}$$

Applications

Template:Central polygonal numbers.svg A fully connected network of n computing devices requires the presence of T_{n − 1} cables or other connections; this is equivalent to the handshake problem mentioned above.

In a tournament format that uses a round-robin group stage, the number of matches that need to be played between n teams is equal to the triangular number T_{n − 1}. For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.