A Digital-Root Property of Consecutive Terms in Geometric Sequences: An Independent Observation, Proof, and Generalization of the 3–6–9 Pattern

This paper presents an independently observed digital-root property of the geometric sequence generated by repeated doubling from 3. For the sequence an=3⋅2na_n = 3 \\cdot 2^n, it is shown that the digital root of the sum of every pair of consecutive terms is always 9. A direct proof is provided using modular arithmetic, followed by a generalization to geometric sequences of the form an=kmna_n = k m^n. The paper derives conditions under which the digital root of consecutive-term sums remains constant, and specifically when it is exactly 9. The phenomenon is further generalized to numeral systems of arbitrary base. The paper also briefly examines the popular association of the numbers 3, 6, and 9 with Nikola Tesla and distinguishes documented mathematics from historically unverified attributions.