# Sum of Arithmetic Series Formula

In the movie "All quiet on the Western front" from 1930 (which is an adaptation of the 1929 novel by Erich Maria Remarque), there is a scene where a soldier at the front mentions the sum of arithmetic series formula. The soldier says
```   S = A + N L/2
```
and finds it beautiful. How does this translate? It deals with the series
```   S = ∑k=0n-1 a+k d = a n  + d n(n-1)/2
= A + n L/2
```
where A=an and L = d (n-1) is the length of the interval under consideration. The formula as stated has been crafted with a mathematician of taste. As the usual formula is more messy, the short version gives the essential and provides the intuition of a rectangular area A plus the area n L/2 of a triangle.

M4V, Ogg Webm. imdb link. The actual novel does not feature this scene. Mathematics appears however (Warning: the novel is a war novel written by a veteran of WWI).
```Bertinck has a chest wound. After a while a fragment smashes away
his chin, and the same fragment has sufficient force to tear open Leer's
hip. Leer groans as he supports himself on his arm, he bleeds quickly, no
one can help him. Like an emptying tube, after a couple of minutes he
collapses. What use is it to him now that he was such a good
mathematician at school.
```
There were probably many Mathematicians who died in WW I. Examples are Andre Bloch

Oliver Knill, Posted July 5, 2019,