# Sum of Arithmetic Progression/Examples/Sum of j from m to n

## Example of Sum of Arithmetic Progression

 $\displaystyle \sum_{j \mathop = m}^n j$ $=$ $\displaystyle m \left({n - m + 1}\right) + \frac 1 2 \left({n - m}\right) \left({n - m + 1}\right)$ $\displaystyle$ $=$ $\displaystyle \frac {n \left({n + 1}\right)} 2 - \frac {\left({m - 1}\right) m} 2$

## Proof 1

 $\displaystyle \sum_{j \mathop = m}^n j$ $=$ $\displaystyle \sum_{j \mathop = 0}^{n - m} \paren {m + j}$ Translation of Index Variable of Summation $\displaystyle$ $=$ $\displaystyle \paren {n - m + 1} \paren {m + \frac {n - m} 2}$ Sum of Arithmetic Progression $\displaystyle$ $=$ $\displaystyle m \paren {n - m + 1} + \frac 1 2 \paren {n - m} \paren {n - m + 1}$ $\displaystyle$ $=$ $\displaystyle m n - m^2 + m + \frac 1 2 \paren {n^2 - m n + n - m n + m^2 - m}$ $\displaystyle$ $=$ $\displaystyle \frac {m + n^2 + n - m^2 + m} 2$ $\displaystyle$ $=$ $\displaystyle \frac {n \paren {n + 1} } 2 - \frac {m \paren {m - 1} } 2$

$\blacksquare$

## Proof 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