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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.?

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 $\sqrt{2}$?

Ford Circles

Age 16 to 18
Challenge Level
The GeoGebra applet below shows the construction of a sequence of Ford Circles. Ford Circles have centre $\left(\frac{p}{q},\frac1{2q^2}\right)$ and radius $\frac1{2q^2}$, where $\frac{p}{q}$ is a fraction in its simplest form (that is, $p$ and $q$ are coprime integers).

In the interactivity below, you can move the sliders to choose values for $a, b, c$ and $d$. The circles have centres $\left(\frac{a}{c},\frac1{2c^2}\right)$ and $\left(\frac{b}{d},\frac1{2d^2}\right)$, and radii $\frac1{2c^2}$ and $\frac1{2d^2}$.

When the two circles touch, they are coloured in blue.
Explore the interactivity and find some values of $a, b, c$ and $d$ that generate circles that touch each other.

In the problem Farey Neighbours, you are invited to explore the value of $ad-bc$ for two adjacent fractions $\frac bd$ and $\frac ac$ from any Farey Sequence.

Explore the value of $ad-bc$ for the touching circles that you have found. 
What do you notice?

Can you prove that for any touching circles in the interactivity above, $|ad-bc|=1$?
Can you prove that, given two such circles which touch the $x$ axis at $\frac bd$ and $\frac ac$, the circle with centre $\left(\frac{a+b}{c+d},\frac1{2(c+d)^2}\right)$ and radius $\frac1{2(c+d)^2}$ is tangent to both circles?