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# Stringing it Out

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### Triangle Incircle Iteration

### Loopy

### Converging Means

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Age 14 to 16

Challenge Level

The invitation is for you to explore the following transformations and comment on what you find:

T(x) = y

T(y) = xy

This transformation has the following successive effect (starting with x):

x

y

xy

yxy

xyyxy

...

...

R(x) = xy

R(y) = z

R(z) = xy

This transformation has the following successive effect:

x

xy

xyz

xyzxy

xyzxyxyz

...

...

S(w) = wx

S(x) = y

S(y) = wz

S(z) = y

This transformation has the following successive effect:

w

wx

wxy

wxywz

...

...

So what do you notice about this pattern and can you explain why it is occurring?

Can you suggest an appropriate set of transformations for the following elements that would have the same effect?

T(v) = ?

T(w) = ?

T(x) = ?

T(y) = ?

T(z) = ?

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.