Consider the sequence defined recursively:
In order to define a sequence recursively, we need to define a starting point. Let’s consider the sequence we create when we let a0 = 12.
Since a0 is even, we use the bottom rule, and find
that a1 = 6.
Since a1 is even, we use the bottom rule, and find
that a2 = 3.
Since a2 is odd, we use the top rule, and find that
a3
= 10.
Since a3 is even, we use the bottom rule, and find
that a4 = 5.
So, when we start with a0 = 12, the first five terms
of the sequence are:
12,6,3,10,5,...
Now, let’s put your pencil to work! Compute the first 14 terms of the
sequence for each of the
following values of a0.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 1 | |||||||||||||
| 2 | |||||||||||||
| 3 | |||||||||||||
| 4 | |||||||||||||
| 5 | |||||||||||||
| 6 | |||||||||||||
| 7 | |||||||||||||
| 8 | |||||||||||||
| 9 | |||||||||||||
| 10 |
Are you starting to see why this sequence is so interesting? Did you notice a similar pattern in most of the examples you worked? Describe what you observed:
___________________________________________________________________________
For the sequences above that did not reach the 1,4,2,... "loop", are you convinced that they will reach the 1,4,2,... "loop"? Justify your belief.
__________________________________________________________________________________
__________________________________________________________________________________
Now that we’ve started to investigate the problem, let’s go to the internet to do some research. We need to get familiar with the terminology of the problem.
Go to this web site:
http://personal.computrain.nl/eric/wondrous/
You will be reading parts 1 through 4, taking notes and answering questions below as you read. You will find some variation in the terminology and notation. (This is very common with mathematical problems that are still being researched.) For example, where we used a1 above, the Roosendaal web site uses S1.
Reading Questions from Roosendaal web site:
What word is used to describe a sequence that reaches this 1,4,2,... "loop"?_______________________
Is this the same way you have used the word "convergent" before?________________________
In your own words, what is the so-called "strong conjecture"? Do you think the conjecture is correct?
_______________________________________________________________________________
What is the alternate terminology (synonym) for "Glide"?_______________________
**Example to accompany Glide definition:
When we start the sequence with 5, the first term in the sequence that
is less than 5
is a3 = 4, so G(5) = 3.
Find:
G(3) ________
G(4)
G(7)
What is the alternate terminology (synonym) for "Delay"?
**Example to accompany Delay definition:
When we start the sequence with 3, the first term in the sequence that
equals 1
is a7 = 1, so D(3) = 7.
Find:
D(4)
D(5)
D(7)
Keep in mind the mathematical discovery process: observe phenomenon, make a conjecture, prove the theorem. You have observed the phenomenon, you probably arrived at the same conjecture before you read it at the Roosendaal web site. This is where all mathematicians are stuck. We have yet to prove the theorem. To the world of mathematics, this is an annoying piece of unfinished business.
Now it’s time for you to do some investigation. You might already have some questions of your own. Here are some questions you might wish to consider:
This site has a program that will generate the sequence for any a0 you input: