Pavan's Pattern: A Shift Identity for Sums of Consecutive Natural Numbers

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.