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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 $\sqrt{2}$?

Dalmatians

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

Thank you Alex from The Grammar School at Leeds for your solution to this problem.

All the sequences starting from 1 to 91 inclusive are 4 cycles (the first, fifth, ninth terms etc. are all equal) for example the sequence starting from 53 is: 53, 25, 48, 76, 53, 25, 48, 76, .... and so on. Alternate numbers in the sequences add up to 101 or 0.

However this is not a general rule. For example, sequences starting with numbers between 92 and 99 also go into 4-cycles but these 4-cycles start from the second term of the sequence. For example the sequence starting with 92 is: 92, 11, 9, 90, - 9, 11, 9, 90, - 9, 11, ...

The number 0 is a fixed point of this system. The problem can be generalised to apply to all integers (expressed as 10$a + b$ as above) and sequences starting from 101, 202, 303 etc. end up at 0.