Sum of Arithmetic Sequence/Examples/Sum of j from m to n/Proof 1

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 $

$=$

$\ds \frac {n \paren {n + 1} } 2 - \frac {\paren {m - 1} m} 2$

$\ds \sum_{j \mathop = m}^n j$

$=$

$\ds \sum_{j \mathop = 0}^{n - m} \paren {m + j}$

$\ds $

$=$

$\ds \paren {n - m + 1} \paren {m + \frac {n - m} 2}$

$\ds $

$=$

$\ds m \paren {n - m + 1} + \frac 1 2 \paren {n - m} \paren {n - m + 1}$

$\ds $

$=$

$\ds m n - m^2 + m + \frac 1 2 \paren {n^2 - m n + n - m n + m^2 - m}$

$\ds $

$=$

$\ds \frac {m + n^2 + n - m^2 + m} 2$

$\ds $

$=$

$\ds \frac {n \paren {n + 1} } 2 - \frac {m \paren {m - 1} } 2$

$\blacksquare$