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Route to Root

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this sequence. What do you notice? Calculate a few more terms and find the squares of the terms. Can you prove that the special property you notice about this sequence will apply to all the later terms of the sequence? Write down a formula to give an approximation to the cube root of a number and test it for the cube root of 3 and the cube root of 8. How many terms of the sequence do you have to take before you get the cube root of 8 correct to as many decimal places as your calculator will give? What happens when you try this method for fourth roots or fifth roots etc.?

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Rain or Shine

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

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Climbing Powers

$2\wedge 3\wedge 4$ 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 $\sqrt{2}$?


Stage: 4 and 5 Challenge Level: Challenge Level:1

Why do this problem?

This problem gives scope for investigation, spotting patterns, working systematically to cover all cases and making and proving conjectures. It provides an example of the mathematics of dynamical systems. This is an important subject in higher mathematics and, in this problem, learners can work with whole numbers in a simple discrete system to discover for themselves the important concepts of cycles and fixed points.

Possible approach

Ensure that the learners understand how the mapping works then suggest that they choose their own starting numbers and work out their own sequences individually, making notes of anything interesting that they observe. They might need to spend time developing a sensible recording system to prevent confusion with the numbers at each step. After about 10 minutes ask the learners to work in pairs and explain to each other what they have discovered. Then later have a class discusion to compare findings from the whole class.

Key questions

What happens to the sequences?
Will they go on for ever? Why?
What patterns do you notice? Can you explain them?
Do sequences have the same behaviour for ALL 2 digit starting numbers? Why?

Possible extension

Investigate the problem for sequences starting with negative numbers or 3-digit numbers or bigger numbers. In what circumstances might fixed points arise? Can students invent similar systems for themselves?

Possible support

Suggest that students start off with the concrete cases $a=b$ for 2 and 3. Then ask what they expect to happen for 88 and 99. Then try it out. Were they correct?

Happy Numbers is a similar problem which lends itself to investigation using spreadsheets.

For a full discussion of some simple discrete dynamical systems see:

Whole Number Dynamics I
Whole Number Dynamics II
Whole Number Dynamics III
Whole Number Dynamics IV
Whole Number Dynamics V.