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## 'Circles Ad Infinitum' printed from http://nrich.maths.org/

Why do this problem?
Scaling is a key idea in mathematics and

this problem provides good practice in working with this
concept and in summing geometric series. Enjoy the idea of being
able to get a hold on an infinite process in a concrete way.

Possible approach
First ask the learners to work out the radius, circumference and
area of the first three circles. (It is often a good strategy in
problem solving to concentrate on $n=1$ first).

Then ask them to work out the radii of the next few circles; this
will concentrate the thinking on scaling.

Then ask them to to work out circumferences and add them up; this
will concentrate the thinking on summing series.

Key questions
Can you work out the radii of the circles?

What are the scale factors?

If you sum the circumferences what sort of series do you get?

If you sum the areas what sort of series do you get?

Can you sum these series?

Possible extension
Try the problem

Von Koch Curve .

Possible support
Try the problem

Smaller and Smaller.