Theorem for Bi‑Arithmetic Grids 行列双等差网格通项定理

For an \(m\times n\) grid of numbers satisfying:

1. every row is an arithmetic progression (AP) left‑to‑right,
2. every column is an arithmetic progression (AP) top‑to‑bottom,

let \(a_{i,j}\) = the entry in row i, column j.

Then there exist four constants \(A,B,C,D\) such that

\(\boldsymbol{a_{i,j}=A+Bi+Cj+D\,ij}\)

holds for all \(i,j\).

Terminology note: this grid is called a bi‑arithmetic grid（双等差网格）.

行列双等差网格通项定理

设一个 \(m\times n\) 的数阵（网格 / 矩阵）满足两个条件：

1. 每一行从左到右的数都是等差数列；
2. 每一列从上到下的数都是等差数列。

记第 i 行、第 j 列的数为 \(a_{i,j}\)（\(i=\)行号，\(j=\)列号），

则一定存在 4 个常数 \(A,B,C,D\)，使得通项公式：

\(\boldsymbol{a_{i,j}=A+Bi+Cj+D\,ij}\)

对网格内所有位置恒成立。

简单记：行等差 + 列等差 ⇔ 通项是常数 + 行一次项 + 列一次项 + 行列交叉项 ij

The numbers, in order, of each row and the numbers, in order, of each column of a  array of integers form an arithmetic progression of length . The numbers in positions  and  are  and , respectively. What number is in position