Wednesday, January 15, 2014

Sums and squares and cubes

Today I'm going to talk about the formula


It's a really really super awesome formula and if you disagree you're probably the kind of person who prefers kittens to puppies.
This formula is super cute and so are these puppies.
Typically this is proven by mathematical induction. If you haven't seen this proof then you probably haven't learned induction yet. It's either a very standard exercise or the 3-4 sources I learned induction from where atypical.

Anyway if you don't already know induction I recommend deriving the proof as a way to teach yourself the technique. You'll need to remember the formula for the sum of the first n integers and how differences of squares work. 

The proof by induction works nicely but it's not as pleasing (to me at least) as the formula itself. So we're left wondering vaguely if there isn't a better way to prove this. I've heard 2 proofs of this which I think of this as "better than the induction but worse than the formula".

I'll give them below and ask the readers to give me something which "feels right". Alternatively we can just collect as many proofs of this as there are puppies in that picture.

First proof (hat tip to Matt Tai for showing me this one).

The above pic clearly has area 13+23+...+n3

We draw it a line and swap the blue shaded areas with the red shaded ones. These are in spite of my terrible drawing skills the same size. So this new shape (including the red and excluding the blue) still has area 13+23+...+n3

On the other hand it's also a triangle so it's area is given by the formula for areas of triangles. Which works out to exactly what we want. 

Second proof (hat tip to David Lipsky for showing me this one).

For the second proof we begin with a square. Of side length 1+2+...+n. Yeah that's right the area is obviously the left side of the equation.

As seen above in badly executed MS paint.

Next we make further use of MS paint and colour this picture in.

If it's not obvious the giant purple part in the middle is the part isn't meant to be all 1 colour, the important thing is this nested way of drawing the picture. The claim is that the area shaded in colour n has area  n

This is actually not too hard to see. We have 1 nxn square with area n2 moving up we have an n*(n-1) rectangle. Which pairs nicely with the 1*n in the bottum left corner as having area n2. Moving up and in from this pair we find an n*(n-2) and a 2*n which give us our 3rd n2 area. We Then go up on the right hand side and right on the bottum row to get more pairs of rectangles with combined areas of n2 , turns out we have n total such areas.  Completing the proof. 
Personally this proof strikes me as nicer than the other 2 but still not quite as nice as the formula.
So please leave better solutions in the comments below.

1 comment:

  1. See, now, how do you expect me to accept your credibility on anything when you open up with such a blatant falsehood as the insinuation that puppies are better than kittens? I'm afraid this post is just too ludicrous for me to take seriously :(