Geometry of Figurate Numbers and Sums of Powers of Consecutive Natural Numbers. Issue 1 (2nd January 2020)

• Record Type: Journal Article
• Authors: Marko, František; Litvinov, Semyon
• Is Part Of: American Mathematical Monthly. Volume 127:Issue 1(2020)
• Page Start: 4
• Page End: 22
• Publication Date: 2020-01-02
• Subjects: MSC: Primary 11B68 -- Secondary 11B83 -- 52C22; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2020.1671129
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: First, we give a geometric proof of Fermat's fundamental formula for figurate numbers. Then we use geometrical reasoning to derive weighted identities with figurate numbers and observe some of their applications. Next, we utilize figurate numbers to provide a matrix formulation for the closed forms of the sums ∑ i = 1 n i p, p = 1, 2, …, thus generating Bernoulli numbers. Finally, we present a formula—motivated by the inclusion-exclusion principle—for ∑ i = 1 n i p as a linear combination of figurate numbers.