Q:

Big-O complexity for n + n-1 + n-2 + n-3 + (...) + 1

I was wondering.. what is the complexity of an algorithm that starts with n elements (which I run through doing whatever). I take one element off, I do it again.. I take off another element and do it again until I have just one element left.
is it O(n log n)? I can't visualize it...

A:

The famous mathematician Gauss is said to have found a formula for that exact problem when he was in primary school. And as mentioned by @Henry in the comments it is:

Source: Wikipedia 
As work is done for every entry, i.e., O(1) is required for each "item". Hence, the problem is in O(n^2). 
Visualisation (also Wikipedia) can be seen as a half filled square: 

A:

To solve the complexity for O(n+n-1+n-2....n times), we need to use Sum for mathematics formula by see this link
=> n+n+n...n times - (1+2+3...n times)
=> n^2- (n^2+n)/2

Complexity will be 
(n^2-n)/2

