Pavan's Pattern: A Shift Identity for Sums of Consecutive Natural Numbers
Year: 2026
Type: preprint
Abstract: For every natural number n, the sum of the first n natural numbers plus n² equals the sum of the next n natural numbers: S1(n) + n² = S2(n). This note gives three short proofs — a term-by-term argument, a closed-form algebraic argument, and a triangular-number formulation — and shows the identity is the special case a = n of a general shift theorem: translating a block of n consecutive integers upward by a increases its sum by exactly a·n. As a corollary, partitioning the naturals into consecutive blocks of length n yields block-sums in arithmetic progression with common difference n². The equivalent classical identity T(2n) = 2T(n) + n² confirms the result independently.
Author: GAMPALA, PAVAN
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203677353
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