Limits of sequences

This function involves absolute values. To find the slope on the slide use different equations to define the function in different parts of its domain.

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?

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

How much of the square is coloured blue? How will the pattern continue?

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

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?

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

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?

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.