Occurred to me while walking the dog this morning:
AVERAGE VALUE * NUMBER OF TERMS
so, for 1 + 2 + … + n this is (n+1)/2 * n
A lot of times, these are cast in terms of the constant difference, the starting term, and the number of terms, as in
s + (s+d) + (s+2d) + … + (s+(n-1)d)
For deriving, it may be easier to think of it as (FIRST + LAST)/2 * n
(s + (n-1)d/2)*n
Since this is essentially a discrete “integral” of a linear function, it may be easier to formulate it as a quadratic, since you have a category for quadratics, and probably recognize that a quadratic is appropriate, leaving less “bits” of information to memorize:
(d/2)n^2 + (s-d/2)n
This formula seems reasonable, since if n=0, the sum is 0, and for n > 1 and d > 1, larger d or n results in a larger sum.
Now, if you take the derivative with respect to n, you get
dn + s - d/2 = s + (n-1/2)d
This struck me as odd at first: it seems that that increasing the number of terms by 1 adds the number halfway between the last term and the next term, whereas what actually gets added is
s + nd
But the derivative can’t be applied that way to a discrete function. What you’re really getting with s + (n-1/2)d is the slope of the line between one value and the next.
Honestly, what puzzles me the most about this is that you said you use it so often you wish you could memorize it. I’m having trouble thinking of when this comes up, as opposed to an integral. But maybe that’s because I haven’t had a remotely technical or scientific job for the past 9 years.