Limits of sequences

problem
Squareness

Age

16 to 18

Challenge level

The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?
problem
Light blue - dark blue

Age

7 to 11

Challenge level

Investigate the successive areas of light blue in these diagrams.
problem
Diminishing returns

Age

11 to 14

Challenge level

How much of the square is coloured blue? How will the pattern continue?
problem
Litov's mean value theorem

Age

11 to 14

Challenge level

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
problem
Archimedes and numerical roots

Age

14 to 16

Challenge level

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?
problem
Summing geometric progressions

Age

14 to 18

Challenge level

Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?
problem
A swiss sum

Age

16 to 18

Challenge level

Can you use the given image to say something about the sum of an infinite series?
problem
Climbing powers

Age

16 to 18

Challenge level

Does it make any difference how we write powers of powers?