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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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Triangle Incircle Iteration

Keep constructing triangles in the incircle of the previous triangle. What happens?

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Skeleton

Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.

Geomlab

Stage: 3, 4 and 5 Challenge Level: Challenge Level:1

This version is sufficiently advanced to run the examples in this worksheet provided you avoid the section on Turtle Graphics.

To visit the application, click on the link below:

Flex GeomLab
Flex GeomLab