Sum of Arithmetic Sequence/Examples/Sum of j from m to n/Proof 2
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Example of Sum of Arithmetic Sequence
| \(\ds \sum_{j \mathop = m}^n j\) | \(=\) | \(\ds m \paren {n - m + 1} + \frac 1 2 \paren {n - m} \paren {n - m + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {n \paren {n + 1} } 2 - \frac {\paren {m - 1} m} 2\) |
Proof
| \(\ds \sum_{j \mathop = m}^n j\) | \(=\) | \(\ds \sum_{j \mathop = 0}^n j - \sum_{j \mathop = 0}^{m - 1} j\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {n \paren {n + 1} } 2 - \frac {\paren {m - 1} m} 2\) | Closed Form for Triangular Numbers |