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

 $\displaystyle \sum_{j \mathop = m}^n j$ $=$ $\displaystyle m \left({n - m + 1}\right) + \frac 1 2 \left({n - m}\right) \left({n - m + 1}\right)$ $\displaystyle$ $=$ $\displaystyle \frac {n \left({n + 1}\right)} 2 - \frac {\left({m - 1}\right) m} 2$
 $\displaystyle \sum_{j \mathop = m}^n j$ $=$ $\displaystyle \sum_{j \mathop = 0}^n j - \sum_{j \mathop = 0}^{m - 1} j$ $\displaystyle$ $=$ $\displaystyle \frac {n \paren {n + 1} } 2 - \frac {\paren {m - 1} m} 2$ Closed Form for Triangular Numbers