I'll be back

Well this blog has been fun. I need to spend less time on the internet so that I can spend some decent time studying. Point is that this is my last blog post until sometime in the middle of November.

But until then, my loyal followers can feast their eyes on a brilliant little infinite series (ie adding an infinite number of numbers together).

The question was on an early high school maths competition paper that my friend did. He only figured out how to do it years later, but the idea behind it is pretty simple. You get shown an image, and the question is, if the red squares keep reducing in size to infinity, what proportion of the total square do the red parts make up?

The first thing to notice is that an infinite number of things added together don't necessarily sum to infinity. The total red part is obviously less than the entire square. So how can we find the answer?

The 'proper' (by which i really mean more rigorous) way to do it is to use the geometric series, which is fairly simple. However, in order to use it, I'd want to show how it works first. I don't have the time, so I'll show you something a little more intuitive.

Let's divide this into an infinite number of chunks. Each chunk is surrounded by green. The largest chunk is on the right.



So how does this help? The idea with this is that we're turning the hard problem into an easy one we can do. That problem is this... What proportion of the area of the chunk on the right is red? The answer is obviously 1/3. What about the next chunk? Well it's exactly the same as the previous chunk, only smaller. So it's 1/3 of that chunk.

This means that each separate chunk is 1/3 red. If 1/3 of every chunk is red, that means that 1/3 of all of the chunks combined is red. What this means is that the red area makes up 1/3 of the area of the entire square.

Problem solved! Well, once exams are done I'll do something on the geometric series (which is awesome), but until then, this will have to do.

Adios, amigos :) (until November 18, anyway :P)