Figurate numbers are numbers that can be arranged in geometric shapes. They include types like triangular numbers, square numbers, and pentagonal numbers. Each type corresponds to a specific geometric figure, with triangular numbers forming triangles, square numbers forming squares, and so on. The nth figurate number can often be calculated using a specific formula related to its shape. These numbers have been studied since ancient times and are linked to various mathematical concepts. For example, Pythagorean triples relate to triangular numbers, while Gauss famously discovered a formula for the sum of the first n integers, which relates to triangular numbers.