# Sum of Arithmetic Sequence/Examples/Sum of j from m to n/Proof 2

 $\ds \sum_{j \mathop = m}^n j$ $=$ $\ds m \paren {n - m + 1} + \frac 1 2 \paren {n - m} \paren {n - m + 1}$ $\ds$ $=$ $\ds \frac {n \paren {n + 1} } 2 - \frac {\paren {m - 1} m} 2$
 $\ds \sum_{j \mathop = m}^n j$ $=$ $\ds \sum_{j \mathop = 0}^n j - \sum_{j \mathop = 0}^{m - 1} j$ $\ds$ $=$ $\ds \frac {n \paren {n + 1} } 2 - \frac {\paren {m - 1} m} 2$ Closed Form for Triangular Numbers