An expression for 2-dimensional element as a sum of differences of elements?

31 Views Asked by At

For 1-dimension, it's simple.

$a_{n}=a_{1}+\sum_{i=2}^{n}(a_{i}-a_{i-1})$

But what would be the corresponding identity for 2-dimension?

In other words, if we put

$a_{n,m}=a_{1,1}+X$

then how can we express $X$ as an elegant sum of differences of $a_{ij}$ s?

The first thing that comes into mind is of course from (1,1) to (n,m), but I failed to put this into an elegant summation of differences of $a_{ij}$ s.