Closed Form for Polygonal Numbers/Examples

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Examples of Closed Form for Polygonal Numbers

Let $\map P {k, n}$ be the $n$th $k$-gonal number.


The closed-form expression for $\map P {k, n}$ for various $k$ can be expressed as:

\(\displaystyle k = 3: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {3 - 2} n - 3 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {n + 1}\) \(\displaystyle \quad = \frac {n \paren {n + 1} } 2\) Definition of Triangular Number
\(\displaystyle k = 4: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {4 - 2} n - 4 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {2 n - 0}\) \(\displaystyle \quad = n^2\) Definition of Square Number
\(\displaystyle k = 5: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {5 - 2} n - 5 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {3 n - 1}\) \(\displaystyle \quad = \frac {n \paren {3 n - 1} } 2\) Definition of Pentagonal Number
\(\displaystyle k = 6: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {6 - 2} n - 6 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {4 n - 2}\) \(\displaystyle \quad = n \paren {2 n - 1}\) Definition of Hexagonal Number
\(\displaystyle k = 7: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {7 - 2} n - 7 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {5 n - 3}\) \(\displaystyle \quad = \frac {n \paren {5 n - 3} } 2\) Definition of Heptagonal Number
\(\displaystyle k = 8: \ \ \) \(\displaystyle \frac n 2 \paren {\paren {8 - 2} n - 8 + 4}\) \(=\) \(\displaystyle \frac n 2 \paren {6 n - 4}\) \(\displaystyle \quad = n \paren {3 n - 2}\) Definition of Octagonal Number

and so on.


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