Gauss's Trick - Arithmetic Sums

[Johnathanrs]

I can't grasp the underlying process on how this is working.

n/2(f+l) = algorithm sum of all integers
n= number of all integers
f= first integer
l= last integer

Example: 1, 2, 3, 4
4/2(1+4)
2(5) = 10

I know how to do it, but I don't really understand how to actually do it. Am I just too stupid?

Why do I need to split the sum of all integers?
Why am I adding the first + last integer?
Why when I times them together does it work?
How did he create the algorithm for this?

 

[SteamKing]

In order to see how this works, write down the string of integers in two different ways:

 1  2  3  4  5  6  7  8  9  10
10  9  8  7  6  5  4  3  2   1

What do you notice about the sum of each column of numbers?

 

[AlephZero]

What Gauss did (according to the usual story) was to pair off the numbers like this.
Suppose you want to sum the 9 numbers 7 8 9 10 11 12 13 14 15
7 + 15 = 22
8 + 14 = 22
9 + 13 = 22
10 + 12 = 22
11 = 22/2
So the sum = (9/2)(22) = (9/2)(7+15)