• Mohsin Bashir Parray

This paper presents a general algebraic identity for three complex numbers in arithmetic progression and investigates its implications for power sums and polynomial equations. Let x,y and z be three complex numbers in arithmetic progression with common difference 'a'. This note establishes a general identity relating the powers of these numbers through the relation xⁿ+yⁿ=zⁿ+K Explicit expressions are obtained for the correction term K for all positive integers 𝑛. The identity follows from the Binomial Theorem and is valid over the field of complex numbers and this also leads to several interesting applications involving power sums, polynomial equations, arithmetic progressions and Diophantine-type problems. The study highlights the algebraic structure underlying the powers of equally spaced complex numbers and suggests avenues for further investigation, including extensions to longer arithmetic progressions and related polynomial identities. Several applications, special cases, and consequences of the identity are presented.

"Formula for Three Complex Numbers in Arithmetic Progression", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.13, Issue 10, page no.c28-c29, October-2026, Available :http://www.jetir.org/papers/JETIR2610205.pdf

"Formula for Three Complex Numbers in Arithmetic Progression", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.13, Issue 10, page no. ppc28-c29, October-2026, Available at : http://www.jetir.org/papers/JETIR2610205.pdf