You can even simply the thought behind this further by simply taking number 1 to n and listing each number in succession and then below each number list n to 1 (backwards).  Then summing down the row.   You'll notice that each number summed down is simply  n+1.  But because we've counted each column of numbers twice, we need to divide by 2 to "normalize" the results and get the correct answer.  So for example take n = 4.  Then write out:

1 2 3 4
4 3 2 1

And now add from the 4 columns:

1 2 3 4
4 3 2 1
----------
5 5 5 5 

Summing this is 5+5+5+5 or [5 x 4 or 5 x (n-1)] = 20.  Since we've double counted each column, we need to normalize by dividing by 2, which results in 20/2 = 10.

Notice that 1 + 2 + 3 + 4 = 10.