TITLE:
On the Study of Binomial Theorem: Formulation of Conjecture for Odd Binomial Coefficients for Binomials with Indices of n = 2r – 1, Where r∈Z-*
AUTHORS:
Ellvan M. Campos, Ronald S. Decano
KEYWORDS:
Binomial Theorem, Binomial Coefficients, Proof by Exhaustion, Conjecture
JOURNAL NAME:
Open Access Library Journal,
Vol.9 No.5,
May
9,
2022
ABSTRACT: Because of the growing role of Binomial Theorem in various fields in Mathematics, such as in calculus and number theory, and even in societal advancements, such in technology and business, mathematicians continue to explore new developments in this interesting theorem. In this case, this study explored some of the unveiled concepts of Binomial Theorem for further studies, specifically on observing the relationships of the indices of the binomials to obtaining odd-valued binomial coefficients. Using the proof by exhaustion, among various mathematical proofs, a formula for n of the binomial in which the cases are separated and focus on the condition is supported. After a series of exhaustion of data, cases were determined and identified. Among the 64 values tested consecutively from 0 to 63, it was found that only the values of 0, 1, 3, 7, 15, 31, and 63 for n resulted to all odd binomial coefficients. Through trial-and-error process, the general term of the indices was expressed in the form of n = 2r – 1, Where r∈Z-*.