 Polygonal number

For integers $s\ge 3$ and $n\ge 1$, the $n$th $s$-gonal number isThese figurate numbers begin at one, and their successive increments form an arithmetic progression:Summing these increments proves the formula. For $s=3,4,5,6$, it gives triangular numbers, square numbers, pentagonal numbers and hexagonal numbers. For example, the first three pentagonal numbers are $1,5,12$, whose successive increments are four and seven. Increasing the number of polygon sides changes the step between increments while keeping this common quadratic structure.

A polygonal number is a type of figurate number that represents a polygon with a certain number of sides. Polygonal numbers can be categorized based on the number of sides in the polygon. The most common types of polygonal numbers include: 1. **Triangular Numbers**: These are the sums of the first \( n \) natural numbers and can be represented as dots forming an equilateral triangle.