# Sum of Arithmetic Sequence/Examples

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## Examples of Use of Sum of Arithmetic Sequence

### Sum of $j$ from $m$ to $n$

 $\ds \sum_{j \mathop = m}^n j$ $=$ $\ds m \paren {n - m + 1} + \frac 1 2 \paren {n - m} \paren {n - m + 1}$ $\ds$ $=$ $\ds \frac {n \paren {n + 1} } 2 - \frac {\paren {m - 1} m} 2$

### Sum of $i + k \paren {2 + 2 i}$

Let $A_n$ be the arithmetic sequence of $n$ terms defined as:

 $\ds A_n$ $=$ $\ds \sum_{k \mathop = 0}^{n - 1} \paren {a_0 + \paren {2 + 2 i} k}$ $\ds$ $=$ $\ds i + \paren {2 + 3 i} + \paren {4 + 5 i} + \paren {6 + 7 i} + \dotsb + \paren {2 n - 2 + \paren {2 n - 1} i}$

Then:

$A_n = n \paren {n - 1} + n^2 i$