General Distributivity Theorem/Examples/Sum of j from m to n by Sum of k from r to s
Jump to navigation
Jump to search
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 $14$