An ascending, then descending sum (2)

In my inaugural blog post I talked about an epiphany that I had regarding the relation

and a pictorial "proof without words".  The mathematical lesson to learn here is to be mentally flexible and look at things from a variety of ways: vertically, horizontally and diagonally.  After all, that was how Georg Cantor took the bull by the horns by tackling uncountable infinite numbers with his diagonal argument.

Seeing that there is someone with high standards who had done Java applet and even gave an exposition of "proof without words", I decided to step up my game and post an Adobe Flash interactive animation of what I had in mind.  You can even click on the "rotate" button to change your "point of view".



I had some fun learning and doing it in Adobe Flash ActionScript 3. AS3 does things differently but is more consistent that the older versions. However, there are some issues with Flash's timing and updating, which I have not found a work-around. Meanwhile, please be gentle with the applet.

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.