Geometry of Figurate Numbers and Sums of Powers of Consecutive Natural Numbers

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 (Formula presented.) thus generating Bernoulli numbers. Finally, we present a formula—motivated by the inclusion-exclusion principle—for (Formula presented.) as a linear combination of figurate numbers.