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:-
   $1+2+...+(k-1)+k+(k-1)+...+2+1$
= $[1+2+...+k]+\left[\right(k-1)+ ...+2+1]$
= $\frac{(k-1)k}{2}+\frac{k(k+1)}{2}$  = $\frac{(k-1+k+1)k}{2}$
= $\frac{\left(2k\right)k}{2}$  = $k^2$

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.