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:

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

$=$

$\ds n a + n \dfrac 1 2 \paren {n - 1} d$

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

$=$

$\ds \dfrac 1 2 n \paren {2 a + \paren {n - 1} d}$

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.

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Sum of Arithmetic Progression to $n$ Terms: $3.1.9$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Arithmetic Series: $19.1$
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arithmetic series
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Example $4$. $(15)$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): arithmetic progression (arithmetic sequence)
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): arithmetic progression (arithmetic sequence)
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 21$: Series of Constants: Arithmetic Series: $21.1.$