Further Extensions of Sury's Identity

TL;DR

This paper introduces new summation formulas related to Sury's identity, connecting Lucas numbers, Fibonacci numbers, and powers of two, using elementary methods.

Contribution

It presents novel summation formulas derived through elementary techniques, expanding the known extensions of Sury's identity.

Findings

Produced numerous new summation formulas from elementary methods.

Connected Lucas numbers, Fibonacci numbers, and powers of two in new identities.

Extended the existing body of work on Sury's identity with fresh results.

Abstract

The equation commonly known as Sury's identity is a deceptively simple summation formula that connects the Lucas numbers, Fibonacci numbers, and powers of two. Many authors have given extensions and generalizations over the years; in this paper, we take a different approach that allows us to produce a good number of new summation formulas, all from elementary (but non-trivial) methods.