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

Example of General Distributivity Theorem

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

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

Proof

\(\ds \sum_{j \mathop = m}^n \sum_{k \mathop = r}^s j k\)

\(=\)

\(\ds \paren {\sum_{j \mathop = m}^n j} \paren {\sum_{k \mathop = r}^s k}\)

General Distributivity Theorem

\(\ds \)

\(=\)

\(\ds \paren {\frac {n \paren {n + 1} } 2 - \frac {\paren {m - 1} m} 2} \paren {\frac {s \paren {s + 1} } 2 - \frac {\paren {r - 1} r} 2}\)

Sum of $j$ from $m$ to $n$

\(\ds \)

\(=\)

\(\ds \frac 1 4 \paren {n \paren {n + 1} - \paren {m - 1} m} \paren {s \paren {s + 1} - \paren {r - 1} r}\)

simplification

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $14$