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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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Divided Differences

When in 1821 Charles Babbage invented the `Difference Engine' it was intended to take over the work of making mathematical tables by the techniques described in this article.

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A geometry lab crafted in a functional programming language. Ported to Flash from the original java at

Odd One Out

Age 16 to 18 Short Challenge Level:

Some of the odd ones out may be easier to spot than others.

You might first like to generate lots of sets of the random numbers so that you can get a feel for the the patterns in the randomness.

Start off by looking to see what sorts of things the numbers have in common and how they may logically be generated. Once you have a logical method of generation (which is only a guess, of course) you can check to see whether all but one of the numbers fits that method of generation. If you think that you have found an explanation, consider the likelihood of numbers generated by some other method accidentally fitting your pattern.