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Describing Patterns in Sequences

Date: 04/16/2002 at 21:40:29
From: Carrie Mack
Subject: Patterning


My students have just started examining number patterns. They are able 
to identify the number pattern corresponding to a number sequence, but 
are having difficulty explaining it in words. I have to make these 
descriptions as simple as possible, and am having some difficulty 
writing a description for the following:

     1   3    6   10    15 ...

They can identify what's happening (going up by 2, 3, 4, 5, etc.) but 
can't grasp the language. How would you explain it using a general 

I hope this is clear. 
Thanks a bunch!

Date: 04/16/2002 at 23:37:18
From: Doctor Peterson
Subject: Re: Patterning

Hi, Carrie.

I often struggle myself when students ask about sequences, because 
there are many different ways to look at them, and I'm never quite 
sure which will be easiest for a particular student to grasp, or will 
look like the most natural way to see it. Part of the problem, of 
course, is that I see these things from a higher perspective, and am 
too familiar with the concepts. In this case, you are probably aware 
of several ways to state what this sequence is:

    the triangular numbers
    successive partial sums of the series 1+2+3+...
    explicitly: a[n] = n(n+1)/2
    recursively: a[1] = 1, a[n] = a[n-1] + n

The last expresses in rather advanced terms the observation you made. 
But students aren't going to follow that.

I'd like to see you do an experiment with your class, and just let 
them brainstorm ways to describe the pattern. As I've suggested, 
there are many valid ways to state it, and many ways also to approach 
a full description starting with simple observations. It could be a 
useful exercise for them to come up with their own list of ways to 
describe it, and perhaps even a chart of different paths to the 
realization of what this sequence really is. You might challenge them 
to find as many observations as they can about the sequence, and then 
to decide which they would start with if they were discussing it with 
younger students, and which, on the other hand, give the clearest and 
most complete explanation of the nature of the pattern. I suppose 
that's cheating, letting them find the answer to your question for 
you; but it just might teach the subject better than just telling 
them what to say about it.

Here is one route they might find:

We can first look for a recursive pattern, that is, a pattern in the 
way each term relates to the one before it. In this case, we see that 
the difference between successive terms increases constantly:

    terms:       1 3 6 10 15
    differences:  2 3 4  5

Now we might want to clarify just what we mean by "increasing"; how 
are the differences increasing? One way is to look at the second 
difference; what is the difference from one difference to the next? 
We see that it is always 1:

    terms:       1 3 6 10 15
    differences:  2 3 4  5
    second diff:   1 1  1

But that's awfully hard to grasp. We might instead look for a way to 
describe the differences explicitly; it can help here to write down 
the index of each term next to it in order to compare:

    indexes:     1 2 3  4  5
    terms:       1 3 6 10 15
    differences:  2 3 4  5

Now we can see that the difference we add to each term to get the next 
is the index of the next term. That makes it a lot easier to describe 
the pattern: to get term N, add N to the previous term.

That's a perfectly good description of the pattern. But there's 
another direction we could go in. Looking back at the differences, we 
can see

    terms:              1   3   6   10   15
    differences:          2   3   4    5
    cumulative sums:  1 + 2=3
                      1 + 2 + 3=6
                      1 + 2 + 3 + 4=10

So we've turned a recursive pattern into one that generates each term 
as a sum.

I don't know whether this is helpful to you, or even comes near what 
you are really asking; but if it only shows that there are many ways 
to talk about the same pattern, and that that observation itself is 
useful to students, perhaps it can help.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Sequences, Series

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