Pavan Gampala's Pattern: A Novel Observation in Arithmetic Sequences

In this paper, we present a newly observed pattern in the sums of consecutive natural numbers. The pattern demonstrates that the sum of the first nnn natural numbers, when added to the square of nnn, equals the sum of the next nnn natural numbers. This finding introduces a unique relationship within arithmetic sequences, offering a fresh perspective on the properties of natural number summation. The implications of this pattern may extend to various areas of number theory, combinatorics, and mathematical analysis.