![Stringing it Out](/sites/default/files/styles/medium/public/thumbnails/content-03-07-six6-icon.gif?itok=BYRpiT2H)
Iteration
![Stringing it Out](/sites/default/files/styles/medium/public/thumbnails/content-03-07-six6-icon.gif?itok=BYRpiT2H)
![First Forward into Logo 10: Count Up - Count Down](/sites/default/files/styles/medium/public/thumbnails/content-00-03-logo1-icon.gif?itok=ayY7PHd-)
problem
First Forward into Logo 10: Count Up - Count Down
What happens when a procedure calls itself?
![First Forward into Logo 1: Square Five](/sites/default/files/styles/medium/public/thumbnails/content-99-07-logo1-icon.gif?itok=PlLWAoqI)
problem
First Forward into Logo 1: Square Five
A Short introduction to using Logo. This is the first in a twelve part series.
![Converging Means](/sites/default/files/styles/medium/public/thumbnails/content-01-05-six6-icon.jpg?itok=1IApwEfD)
problem
Converging Means
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the two sequences.
![Differs](/sites/default/files/styles/medium/public/thumbnails/content-00-05-six3-icon.jpg?itok=En5rHzFr)
problem
Differs
Choose any 4 whole numbers and take the difference between
consecutive numbers, ending with the difference between the first
and the last numbers. What happens when you repeat this process
over and over again?
![Loopy](/sites/default/files/styles/medium/public/thumbnails/content-03-06-15plus1-icon.gif?itok=kDuttfvk)
problem
Loopy
Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$
for different choices of the first two terms. Make a conjecture
about the behaviour of these sequences. Can you prove your
conjecture?
![Triangle Incircle Iteration](/sites/default/files/styles/medium/public/thumbnails/content-00-12-15plus5-icon.jpg?itok=F--zOF-o)
problem
Triangle Incircle Iteration
Keep constructing triangles in the incircle of the previous triangle. What happens?
![Climbing Powers](/sites/default/files/styles/medium/public/thumbnails/content-00-09-15plus1-icon.jpg?itok=pPouFKID)
problem
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 $\sqrt{2}$?
![Rain or Shine](/sites/default/files/styles/medium/public/thumbnails/content-00-05-15plus3-icon.jpg?itok=nsWX66Jx)
problem
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.
![Route to Root](/sites/default/files/styles/medium/public/thumbnails/content-00-03-15plus1-icon.jpg?itok=bT2LivdM)
problem
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.?