Working with Sequences

Date: 11/14/98 at 09:16:12
From: Kathy Kwan
Subject: Sequences

I have tried to do the following questions twice, but I still couldn't 
figure them out. Would you please show me how to do them?

1) Give the next two terms of the sequence:

   1, 1, 2, 4, 3, 9, 4, ...

2) Write down the first four terms and the seventh term of the 
sequence for which the nth term is given.

   a) 2n - 1  
   b) n - 4

Date: 11/19/98 at 21:20:46
From: Doctor Anderson
Subject: Re: Sequences

Hi, Kathy.

These can be tricky, especially the first one, where you have to find 
a pattern. You can think about it for hours, trying a million 
different patterns, and never get the right one. 

I don't know about you, but my first instinct on question (1) is to 
figure out what to add to or subtract from each term to get the next 
term. This is often a good method, and we can sometimes find the answer 
easily by writing out this sequence: 

   1-1, 2-1, 4-2, 3-4, 9-3, 4-9, ...

which is the same as:

   0, 1, 2, -1, 6, -5,...

So to get the second term, add 0 to the first, to get the third, add 1 
to  the second, to get the fourth, add 2 to the third, to get the 
fifth, add (-1) to the fourth, etc. Well, I don't see a nice pattern, 
do you?  So although this often works, it doesn't seem to here. 

Let's think of another way to look at it. I can't really help you here 
without telling you the right way to go. I can't give you the actual 
answer, but here is the right direction on this one. Pair up terms 
that are next to each other. Do it like this: (1,1), (2,4), (3,9), 
and you can't make the next pair yet, call it (4,x). Look at these, 
especially (2,4) and (3,9), for a while. What kind of relation does the 
second number have with the first, in each pair? This should help you 
find x, once you see the pattern. Now, that gives you the next term, so 
what about the one after that? Well, look at the sequence of first 
terms of each pair. This has a really simple pattern. So far we have 
(1,1), (2,4), (3,9), and (4,x), so let's call the next pair (y,z). I 
hope you can find y without too much trouble, and you don't even need 
to find z (the question doesn't ask for it).

Now, for question (2). I will solve an example like yours, all the way 
through. 

Let's use the sequence with the nth term being 3n+2. What is another 
way of saying the first term? It is the term where n = 1. Well, if 
n = 1, then 3n + 2 = 3(1) + 2 = 5. So we can make a simple chart 
following this procedure:

   n    3n+2
   ------------------
   1    3(1) + 2 = 5
   2    3(2) + 2 = 8
   3    3(3) + 2 = 11
   4    3(4) + 2 = 14
   7    3(7) + 2 = 23

From here, you should be able to find the first 4 terms and the 7th 
term of your problems. Good luck, and if you get stuck, feel free to 
ask for more help.

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