Sequences

Teacher Resources

[General teacher guide]

The Number Sequences Activity Sequence document can be downloaded here

Related worksheets

Exploring basic sequences

This set of tasks is the starting point for anyone who intends to participate in the sequences activites. It requires very little prior experience in ToonTalk - a few hours of playing the puzzle game will do.

Each task is presented as a task-template. Go to the task page, and click the "new report from this template" button. This will lead you to the form for creating a new web-report. After you have done so, make sure you click the "save" button on your new report. Now follow the insturctions, and edit your report as you progress in the task.

The tasks are:

After you've completed those, you can try this task:

• Add Up surprises

Guess my Robot

Guess my Robot is a game in which proposers train a robot to generate a sequence, and responders try to guess that robot by looking at the first few terms it generated.
To participate in the game, you should first read:

• The Rules of the Game.

Then, choose a challenge to respond to from:

• The main game page
  

When you feel you're up to it, you can pose your own challenge using:

• The Guess my Robot Template

And write about yours and others' sequences

• My favourite sequence
• Reflections on Guess my Robot

You can also try the related:

• Small change challenge

Learning snapshots and Pedagogical advice

• Learning snapshot: Guess my robot, the case of Joe 999
  How one boy used ToonTalk to make a mathematical argument.
• Learning snapshot: Online collaboration in Cyprus
  Cypriot students reflect on sharing robots, and what they learnt from it.
  
• Pedagogical advice: Canonical sequences
  Mathematical structure of common student sequences.
• Pedagogical advice: On teaching number sequences
  How to discuss sequences in the classroom.

General background

A word document version of this report is available here Sequences_Guidance.doc

Introduction

1. Definition of a sequence

2. Example 1 - Graph of a sequence

3. Finding the general term

4. Differences of Sequences

The approach of using differences of terms does not help for all the problems involving sequences. However, for many sequences, the use of differences can be very profitable and, moreover, the differences themselves prove extremely valuable in many other contexts. For our considerations it will be useful to denote the difference between two successive terms of a sequence by means of the difference operator , which is defined for any sequence A = {a₁, a₂,…, a_n …}, as follows:
    a₁= a₂ - a₁,
    a₂= a₃ - a₂,
and, in general
    a_n = a_n+1 - a_n
for any value of n. The difference operator represents the change in the sequence. A useful application of this operator arises when we apply it to a sequence of terms arising from a linear function.

4.1 Example

n	an	an
1	-2	3
2	1	3
3	4	3
4	7	3
5	10	3
6	13	3

Notice that the first six terms are the same as those in Example 1 and the differences a_n are all constant.  The value of this constant is 3, which is exactly the slope of the graph of the linear function a_n =3n - 5. This is not an accident, as Theorem 1 shows.

Theorem 1. If a_n = cn + b (where c and b are constants and this holds for n = 1, 2, 3, …), then

Proof
1.  a_n = a_n+1 - a_n =  (c(n+1) + b) - (cn + b) = c
2. Successive points on the line are (n, a_n) and (n + 1, a_n+1). The slope m of the segment connecting these points is obtained in the usual way as the difference in y-coordinates divided by the difference in x-coordinates. Therefore:
m = (a_n+1 - a_n ) / (n + 1 - n) =  c (n + 1) + b - (cn + b) = c
It would be interesting to know if the converse of Theorem 1 holds. If we know that the differences of the sequence {a_n} are constant  for all n, can we conclude that there are constants c and b so that a_n = cn +b for all n? The answer is yes, as seen in Theorem 2.

Theorem 2. If a_n = c (where c is a constant independent of n) then there is a linear function for a_n (i.e. there exists a constant b so that a_n = cn +b ).
To see how to prove this theorem, try taking a specific sequence with a constant difference and try working out a formula for a_n. This will give you the main idea.
How could we apply Theorem 2? Suppose we had started with the values in the table but didn't know the function from which the data came. We could reconstruct this function in the following way. The fact that the differences are all 3 shows that we want a formula of the form a_n = 3n +b. To find b, we will use the fact that a₁ =  2. Thus,
-2 = 3.1 + b, so b = -5. Therefore, our function is
a_n =3n + 5

4.2 Example

Let us consider the sequence
    A = {1, 3, 6, 10, 15, 21,  …}.
If we apply the difference operator to the elements of the sequence, we obtain a new sequence of differences:
    A  = {2, 3, 4, 5, 6,  …}.
Since this itself is a sequence of numbers, we can apply the difference operator to it. Let us denote  (a_n) by ² a_n and call it the second difference.
We find in this example that all the second differences are equal to 1, or equivalently,
   ² A  = {1, 1, 1, 1,  …}.

5. Behaviour of Sequences

What does it mean for all the second differences to be constant, as in the above example? In particular, it might indicate something about the function from which the data values originally came. The first difference tells us the difference in successive values for a sequence. The larger the first difference is, the faster the sequence grows. When the first difference is relatively small but positive, the sequence is growing more slowly. When the first difference is negative, the sequence is decreasing. When the first difference is constant, the sequence is changing at a constant rate. In general, the first difference measures "change" of the sequence.
Now suppose that the second difference is a positive constant, and the first difference is positive. This tells us that the rate of growth is growing, i.e., the sequence is growing even faster.  Consequently, we see that the original sequence cannot be defined as a linear function, because linear functions grow at a constant rate and have a second difference equal to 0. Thus when the second differences are positive, the sequence has to grow faster than a linear function. While there are many possible candidates for such a function, the two most prominent are exponential functions and polynomials.

6. Number sequences for physics, engineering and art

Sequences of numbers have unexpected and practical uses in many areas of science and engineering, including acoustics. For example they find application in measuring concert hall acoustics, radar echoes from planets, the travel times of deep-ocean sound waves for monitoring ocean temperature, and improving synthetic speech and the sounds associated with computer music.

7. Sources cited

Table of all Available Teacher Resources