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