You may wish to take a look at Diminishing Returns before working on this problem.
In the interactive environment below, you can move the slider to see how the image is built up.
At each stage of the process, what proportion of the image is coloured blue?
What would happen if we continued the process forever?
Once you've had a chance to think about these questions, click below to see some different approaches to understanding the process.
Below is a collection of images created using repeating processes.
For each pattern, can you find a sequence of fractions whose sum represents the blue area?
For each pattern, can you find different ways to work out the total proportion shaded blue?
You may also be interested in the other problems in our Journeys of Discovery Feature.