+-+-+-+ . . .+-+-+-+-+-+-+ -
| |1|2| |n|n|n|n|n|n| |
+ +-+ + +|+|+|+|+|+ + |
| | | | |5|4|3|2|1| | |
+ + +-+ + + + + + + + |
. |
. |
. |
+ + + + + + + + + + + |
| | | | | | | | | | | |
+ + + + +-+ + + + + + n
| | | | | | | | | | | |
+ + + + + +-+ + + + + |
| | | | | | | | | | | |
+ + + + + + +-+ + + + |
| | | | | | | | | | | |
+ + + + + + + +-+ + + |
| |n|n| | | | | | | | |
+ +|+|+ + + + + +-+ + |
|n|1|2| |5|4|3|2|1| | |
+-+-+-+ . . .+-+-+-+-+-+-+ -
|-------- n+1 -----------|
Therefore, 1+2+3+4+5+...+(n-2)+(n-1)+n = n(n+1)/2.
For example:
######********************* ---
######********************* |
## #* 1 ** 2 ** 3 ** 4 ** |
## #* ** ** ** ** |
## #****** ** ** ** |
## ######* ** ** ** |
## ## #* ** ** ** |
## ## #* ** ** ** |
## ## #****** ** ** 4
## ## ######* ** ** |
## ## ## #* ** ** |
## ## ## #* ** ** |
## ## ## #****** ** |
## ## ## ######* ** |
## ## ## ## #* ** |
## 4 ## 3 ## 2 ## 1 #* ** |
#####################****** |
#####################****** ---
| |
|------------5------------|
| |
Therefore, 1+2+3+4 = 4*5/2 = 10.

There are, of course, other ways to prove what 1+2+3+4+ . . . +(n-1)+n equals, but IMHO, a proof by picture is the most satisfying.