A pentagonal number is a figurate number that represents a pentagon. It can be calculated using the formula P_n = \fracn(3n - 1)2 , where n is a positive integer. The sequence starts with 1, 5, 12, 22, and so on, each representing the total number of dots that can form a pentagon. These numbers are part of a broader category known as polygonal numbers, which include triangular and square numbers. Pentagonal numbers have applications in various areas of mathematics, including combinatorics and number theory, where they help in understanding patterns and relationships among numbers.