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