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

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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 1

\(\ds \sum_{j \mathop = m}^n j\) \(=\) \(\ds \sum_{j \mathop = 0}^{n - m} \paren {m + j}\) Translation of Index Variable of Summation
\(\ds \) \(=\) \(\ds \paren {n - m + 1} \paren {m + \frac {n - m} 2}\) Sum of Arithmetic Sequence
\(\ds \) \(=\) \(\ds m \paren {n - m + 1} + \frac 1 2 \paren {n - m} \paren {n - m + 1}\)
\(\ds \) \(=\) \(\ds m n - m^2 + m + \frac 1 2 \paren {n^2 - m n + n - m n + m^2 - m}\)
\(\ds \) \(=\) \(\ds \frac {m + n^2 + n - m^2 + m} 2\)
\(\ds \) \(=\) \(\ds \frac {n \paren {n + 1} } 2 - \frac {m \paren {m - 1} } 2\)

$\blacksquare$


Proof 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


Sources

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