The Concept of Square Diagonal Constants and Its Application to Arithmetic Series

When consecutive numbers, or the terms of an arithmetic progression, are written row by row in an n × n square, the sums across the two diagonals are equal. This note records that observation and shows that the diagonal sum has the form na + k, where a is the first term and k depends only on the side n. The constant k is named the Square Diagonal Constant (SDC), and a dedicated notation is introduced for it. For first term a and common difference d, the diagonal sum is na + d(n³ − n)/2, and the sum of all the numbers in the square is n times this. The note gives a proof, worked examples (including a large-scale problem), and a method for summing the first x terms of an arithmetic progression by fitting them into a square of side √x. This is exploratory student work. The same formula is known in the magic-square literature (Hunter and Madachy, 1975), and this is stated in the paper. The author’s original handwritten working is included as Appendix A. An AI model (Claude, Anthropic) was used to generate the LaTeX code and formatting and to assist with the wording and written proofs, as acknowledged in the paper.