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Triangle Incircle Iteration

Keep constructing triangles in the incircle of the previous triangle. What happens?

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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?

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Converging Means

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.

Stretching Fractions

Age 14 to 16 Challenge Level:
What is the routine for calculating the result when the fraction is less than or equal to a half ?

What is the routine when the fraction is greater than a half?

Can you express that in algebra?

If $x$ is the fraction value what is the result in terms of $x$ for the two cases above?

For each denominator (bottom number of the fraction) which fractions are equivalent to other fractions with a smaller denominator?

Which fractions form a loop?

Which fractions do not belong to the loop but track into it?