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Triangle Incircle Iteration

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?

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.

Peaches in General

Age 14 to 16 Challenge Level:

Why do this problem?

This an excellent problem through which to give students an opportunity to experience iterative processes. It also moves, by easy stages, into a multi-variable problem, and the value of using a spreadsheet is readily apparent.

Possible approach

Starting with the initial given problem discuss solutions found by the group.

Invite ideas about possible directions for generalisation, perhaps starting with the easier results like allowing 'plus one more' to become plus two, plus three, and so on.

Clarify what 'result' has actually been discovered for each generalisation and spend plenty of time letting students sense the 'mathematical need' to account for each 'result'. These are good questions to be 'left in the air', allowing students to turn these over in their minds over time.

Key questions

  • What generalisations are possible ?
  • If we explore those one at a time, which shall we take first ?
  • What general statement can we now make ?
  • Can we justify that ? Explaining why it should be so.

Possible extension

More able students will produce more extended generalisations and have a motivation to account for what is observed, challenging one another to communicate clear explanations or visualisations of the fundamental processes.

Able students will sense the potential power of a spreadsheet and should be encouraged to work collaboratively to become proficient and confident in its use.

Possible support

The Stage 3 problem is sufficient as it is, to offer less able students a rich experience producing results that form a clear pattern which will not have been obvious at the start.