Sum of Arithmetic Progression/Examples

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Examples of Use of Sum of Arithmetic Progression

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

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

Sum of $i + k \left({2 + 2 i}\right)$

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

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

Then:

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