The Four-Operation Successor-Square Pattern

This mathematical note presents an elementary four-operation construction in which addition, subtraction, multiplication, and division applied to the same non-zero number produce, after summation, the square of its successor. The relationship is proved algebraically and examined for positive integers, negative numbers, and fractions. The note distinguishes the independently observed arithmetic formulation from the classical algebraic identity (n+1)^2 = n^2 + 2n + 1. It also discusses the restriction n ≠ 0, the special case of zero, and possible directions for generalization. The purpose of this note is to document the observation, establish its general mathematical form, provide an algebraic proof, and illustrate how a simple numerical experiment can lead to a general mathematical statement.