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

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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

$\blacksquare$


Sources