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When meeting geometric series for the first time, there's a temptation to look at the algebraic definitions first and derive the summation formula without giving students the opportunity to get to grips with why some sums converge and some diverge. In this problem, by offering a geometrical representation, we hope it will be clear to students why these series converge, while providing a hook to inspire curiosity into exploring other convergent series.
What fraction of the square is blue at each stage?
How can you visualise the pattern to help you to find the limit?
The patterns could be used as a brief introduction to the idea of self-similarity in fractal patterns.
There are suggestions in the Teachers' Resources to Diminishing Returns which might be useful to support students who are struggling with this task.