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### Terms and Rules

```
Date: 12/14/2001 at 09:55:30
From: William Johnson
Subject: Terms and rules

My step daughter is in sixth grade and she has been doing a pattern
journal where she has two columns of numbers: the first column is the
n, and the second colum is the term, and she has to find the rule
(e.g. n2), etc. Is there some place I can find out the basics for this
concept?
```

```
Date: 12/14/2001 at 23:17:02
From: Doctor Peterson
Subject: Re: Terms and rules

Hi, William.

problem. A lot depends on the level of difficulty; most likely she has
been given a set of problems that all have a similar kind of pattern,
so that a method can be developed for solving them. The problems can
differ in the presentation (whether consecutive terms are given, for
example), and in the complexity of the rule (a simple multiplication,
a more complicated calculation from n, a recursive rule - based on the
previous term - or even a weird trick rule like "the number of letters
in the English word for the number n"). For that reason, it would be
very helpful if you could send us a couple sample questions so we
could help more specifically.

I don't see any good examples in our archives at the elementary level
- probably just because we've never felt that any one such answer was
useful to help others. Let's try a few samples. Here's an easy one:

n | term
---+------
1 | 3
2 | 6
3 | 9
4 | 12

Here you may just be able to see that the terms are a column in a
multiplication table (or, more simply, that all the terms are
multiples of 3); or you might look at the differences between
successive terms (a very useful method at a higher level, called
"finite differences") and see that the terms are "skip-counting" by
3's, a clue that the rule involves multiplication by 3. However you
see it, the rule (using "*" for the multiplication sign) is

3 * n

Here's a slightly more complicated one:

n | term
---+------
1 | 6
2 | 11
3 | 16
4 | 21

Here the differences are all 5, suggesting multiplication by 5. You
might want to add a new column to the table so you can compare the
given term with 5n:

n | 5n | term
---+----+------
1 | 5  | 6
2 | 10 | 11
3 | 15 | 16
4 | 20 | 21

Now you can see that each term is one more than 5n, so the rule is

5n + 1

Now they might get still more complicated:

n | term
---+------
1 | 2
2 | 5
3 | 10
4 | 17

Here the differences aren't all the same (3, 5, 7), so something
besides multiplication and addition must be going on. I happen to know
that when the differences are themselves increasing regularly (by 2
each time in this case) that there is a square involved; try adding a
column for the square of n and see if you can figure it out.

This just gives a small taste of what these puzzles might be like.
They can get much harder. Actually, in many harder cases I think the
problems can be very unfair, because in reality (when you don't know
ahead of time what kind of rule to expect) the rule could be
absolutely anything, such as "the nth number on a page of random
numbers I found in a book"! It's really just guesswork, and sometimes
it's really hard to say which of several possible answers is what the
author of the problem might have had in mind. But at your daughter's
level you don't have to worry about that; it may be mostly just a
matter of recognizing familiar patterns such as skip counting or
squares. It's only where I sit, seeing problems completely out of
context, and having to make a wild guess as to what sort of rule to
look for, that these problems are always challenging.

If your problems aren't like these, and you need more help, feel free
to show them to me, so I can suggest ways to handle them.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 12/15/2001 at 13:39:31
From: William Johnson
Subject: Terms and rules

Thank you, Dr. Peterson. That is exactly what I was talking about and
you gave me great examples. One question I do remember was connected
pentagons 1 = 5, sides -2 = 8 - 3 = 11 etc. where n is the number of
pentagons and the term is the number of sides exposed, and then we
helped.

Thanks,
William F. Johnson
```

```
Date: 12/15/2001 at 22:30:49
From: Doctor Peterson
Subject: Re: Terms and rules

Hi, William.

My answer is most helpful if you are just given a list of terms, and
have to guess the rule. Your example is actually easier in some
respects, and harder in others. It's common to just make a table of
terms based on the geometry of the problem, and then try to guess a
rule from that. But how do you know that the rule is really the right
one, when it's not just an arbitrary list of terms, but one generated
by a real situation? You really need to find some reason for
connecting the rule to the objects you are counting. For that reason,
I recommend trying to find a rule in the counting process itself.

How do you count the exposed sides? You start with 5; then when you
add a second pentagon, you add 5 more sides, but one side from each
pentagon becomes "hidden"; so the new number is 5 + (5 - 2) = 8. When
you add another (presumably always on the "opposite" side of the
pentagon, to avoid overlapping with those already placed), you again
add 5 and take away 2; so the rule seems to be that you start with 5
and add 3 for each pentagon after the first. This leads directly to
the rule, with no need to guess.

So you can use the guessing approach if you want, and I suspect many
texts and teachers expect that, but it's much more satisfying to skip
the tables and really know you have the right answer!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Puzzles

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