The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?
An article introducing continued fractions with some simple puzzles for the reader.
Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?