+-+-+-+ . . .+-+-+-+-+-+-+   -
| |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.

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