Sum of Arithmetic Sequence/Also presented as

Sum of Arithmetic Sequence: Also presented as
The Sum of Arithmetic Sequence can also be seen presented in the forms:


 * $\ds \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} = n a + n \dfrac 1 2 \paren {n - 1} d$


 * $\ds \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} = \dfrac 1 2 n \paren {2 a + \paren {n - 1} d}$

Some present it as:

$\ds \sum_{k \mathop = 0}^n \paren {a + k d} = a \paren {n + 1} + \dfrac 1 2 d n \paren {n + 1}$

The reality is that this is a messy result that cannot be presented elegantly.