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.?

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.

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 $\sqrt{2}$?

Dalmatians

Stage: 4 and 5 Challenge Level:

Investigate the sequences obtained by starting with any positive 2 digit number $(10a+b)$ and repeatedly using the rule

$10a + b \to 10b -a$

to get the next number in the sequence.

NOTES AND BACKGROUND

You can take any number and write it in the form $10a+b$ , that is as a multiple of ten plus a number $b$ between 0 and 9, for example:

$$57 = 10 \times 5 + 7\quad\quad -6 = 10 \times (-1) + 4 \quad\quad 123 = 10\times 12 + 3$$

This iterative procedure is an example of a dynamical system which can be studied in more detail at university; you may read an introduction to this fascinating subject in Whole Number Dynamics 1 . Dynamical systems using decimals can have many strange and interesting properties; they form the foundation of the subject of chaos, which you can read about on the Plus website .