An Elementary Proof of the Generalization of the Binet Formula for $k$-bonacci Numbers

TL;DR

This paper provides an elementary proof for the generalized Binet formula applicable to $k$-bonacci numbers, offering a straightforward way to compute these sequences using roots of their characteristic polynomial.

Contribution

It introduces a simple, elementary proof of the generalized Binet formula for $k$-bonacci numbers, expanding understanding of their closed-form expression.

Findings

Elementary proof of the generalized Binet formula

Closed-form calculation for $k$-bonacci numbers

Utilizes roots of characteristic polynomial

Abstract

We present an elementary proof of the generalization of the $k$-bonacci Binet formula, a closed form calculation of the $k$-bonacci numbers using the roots of the characteristic polynomial of the $k$-bonacci recursion.