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To iterate means to repeat, so an iterative process involves repeating something many times.
Imagine some dough, Plasticine or Blu-Tack, something that can be made into a strip then stretched.
We are going to take a length of this material, which we'll regard as the unit length, and put a mark at some fraction distance along it.
Now we are going to follow a procedure and see where our mark ends up.
The material is folded in the middle so that the bottom reaches back to the top.
The material is now only half a unit in length and twice as fat, so it is rolled out or stretched uniformly to become one unit in length again.
Finally we'll note the new position of our mark.
What happens for other start fractions?
Does everything go to a loop?
What size loops appear and for what fractions?
Keep constructing triangles in the incircle of the previous triangle. What happens?
Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the two sequences.