An ascending, then descending sum

While tutoring lower secondary olympiad maths, my student and I encountered an interesting sum

The numbers start from 1 and go up to a maximum value $k$, and then go down back to 1. If one knows summation formulas, there is no problem with evaluating this sum.  The answer is $k^2$.

Using $1 + 2 + ... + k = \frac{k(k+1)}{2}$ (provable using Gauss' trick), we can proceed as follows:-
   $1 + 2 + ... + (k-1) + k + (k-1) + ... + 2 + 1 $
= $[1 + 2 + ... + k] + [(k-1) +  ... + 2 + 1] $
= $ \frac{(k - 1) k}{2} + \frac{k(k+1)}{2} $  = $ \frac{(k - 1 + k + 1) k}{2} $
= $ \frac{(2 k ) k}{2} $  = $ k^2 $

Straightforward exercise for the left-brain, especially for older kids in Junior College.  But ... where is the insight?  Thanks to Descartes, Mathematics is as much a right-brained activity as a left-brained one.


Then my right brain had an epiphany ("ting!" with flashing lightbulbs).  I realised that there is a way to visualise this fact and got the "Hey!  I never looked at it this way" kind of feeling.


Challenge to the reader: Is there a way to visualise this sum?  Can you do a "Proof without words"?

Think.  Then refer to here.