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Describing Patterns in Sequences

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

Hi,

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
description?

I hope this is clear.
Thanks a bunch!
Carrie
```

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

Hi, Carrie.

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

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
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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

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