General Distributivity Theorem/Examples

Example of General Distributivity Theorem

Sum of $j$ from $m$ to $n$ by Sum of $k$ from $r$ to $s$

$\ds \sum_{j \mathop = m}^n \sum_{k \mathop = r}^s j k = \frac 1 4 \paren {n \paren {n + 1} - \paren {m - 1} m} \paren {s \paren {s + 1} - \paren {r - 1} r}$

for $m \le n, r \le s$.