Combinatorics and Graph Theory

A Study on the Derivation of General Terms and Summation Formulas for Difference Sequences Using the General Term of Arithmetic Sequences and Their Generalisation to Higher-Order Difference Sequences.

Authors: Hisanori Sakamoto
Comments: 11 Pages. In Japanese

This research presents a novel method of deriving the general term of difference sequences that does not use sigma notation. By treating the first and second differences as arithmetic sequences and focusing on the relationships between terms, we developed a direct formula that connects the initial terms and differences. From this formula, we derived a new summation formula with coefficients displaying an elegant pattern of factorial reciprocals. We then generalised this approach to higher-order difference sequences, expressing the general term using binomial coefficients. This method provides an intuitive understanding of difference sequences using only basic sequence knowledge.
Category: Combinatorics and Graph Theory