# General Distributivity Theorem/Examples/Sum of j from m to n by Sum of k from r to s

## Example of General Distributivity Theorem

$\displaystyle \sum_{j \mathop = m}^n \sum_{k \mathop = r}^s j k = \frac 1 4 \left({n \left({n + 1}\right) - \left({m - 1}\right) m}\right) \left({s \left({s + 1}\right) - \left({r - 1}\right) r}\right)$

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

## Proof

 $\displaystyle \sum_{j \mathop = m}^n \sum_{k \mathop = r}^s j k$ $=$ $\displaystyle \left({\sum_{j \mathop = m}^n j}\right) \left({\sum_{k \mathop = r}^s k}\right)$ General Distributivity Theorem $\displaystyle$ $=$ $\displaystyle \left({\frac {n \left({n + 1}\right)} 2 - \frac {\left({m - 1}\right) m} 2}\right) \left({\frac {s \left({s + 1}\right)} 2 - \frac {\left({r - 1}\right) r} 2}\right)$ Sum of $j$ from $m$ to $n$ $\displaystyle$ $=$ $\displaystyle \frac 1 4 \left({n \left({n + 1}\right) - \left({m - 1}\right) m}\right) \left({s \left({s + 1}\right) - \left({r - 1}\right) r}\right)$ simplification

$\blacksquare$