>>12824500A visual proof:
You can rewrite the sum as 2 (1+2+...+n) - n
Create a n by n grid. It has n^2 elements.
The upper triangular part has 1 + 2 + ... + n elements.
The same goes for the lower triangular part.
The diagonal has n elements.
The upper and lower triangular parts comprise the whole grid and they overlap only on the diagonal.
So we have:
square grid = upper triangular part + lower triangular part - diagonal
n^2 = (1 + 2 + ... + n) + (1 + 2 + ... + n) - n
