Common Difference, Common Ratio

Date: 04/21/2003 at 18:54:30
From: Bobby
Subject: Identify the common difference in each arithmetic sequence

2, 6, 10, 14

Date: 04/21/2003 at 21:04:50
From: Doctor Achilles
Subject: Re: Identify the common difference in each arithmetic 
sequence

Hi Bobby,

Thanks for writing to Dr. Math.

First let's define the term "arithmetic sequence" so we're both on 
the same page. An arithmetic sequence is made by choosing two 
numbers: the starting number and the "common difference." The 
starting number is the first number in your sequence, and you then 
make each additional number by adding your common difference to the 
term before.

That's a bit complicated sounding, so let's try a couple of 
examples.  Let's make an arithmetic sequence with a starting number 
of 3 and a common difference of 5.

Our first term will be our starting number:

  3

Our second term will be equal to the first term (3) plus the common 
difference (5); so our second term is 8. So the first two terms of 
our sequence are:

  3, 8

Our third term will be equal to the second term (8) plus the common 
difference (5); so our third term is 13. So the first three terms of 
our sequence are:

  3, 8, 13

Our fourth term will be equal to the third term (13) plus the common 
difference (5); so our fourth term is 18. So the first four terms 
of our sequence are:

  3, 8, 13, 18

Do you see how the sequence grows from there?

Try finding the first five terms in an arithmetic sequence that has 
a starting number of 7 and a common difference of 6. Let me know if 
you get stuck doing this.

Now that you're familar with making an arithmetic sequence from a 
starting number and a common difference, the problem you're doing is 
asking you to find the common difference of a given sequence.

The first term in your sequence is 2; so that is your starting 
number.

The second term is 6. To find the difference between this and the 
first term, we take 6 minus 2; that gives us 4. So the difference 
between the first and second terms is 4.

The second term is 6 and the third term is 10. To find the 
difference, we take 10 minus 6; that gives us 4 again.

So the common difference between each term is 4.

That is all the information you need to solve this type of problem.  

If you would like to know a little more about sequences, then read 
on but this is not relevant to arithmetic sequences.

There is one other common type of sequence called a geometric 
sequence. Geometric sequences also have a starting number, just like 
arithmetic sequences. However, they do not have a common difference; 
instead they have what is called a "common ratio." The common ratio 
is what you multiply each term by to generate the next term in the 
sequence.

For example, you could have a starting number of 4 and a common ratio 
of 3.

Your first term will be the starting number. So the first term is:

  4

Your second term is going to be equal to the first term (4) 
multiplied by the common ratio (3); this equals 12. So your second 
term is 12, and the first two terms are:

  4, 12

Your third term is going to be equal to the second term (12) 
multiplied by the common ratio (3); this equals 36. So your third 
term is 36, and the first three terms are:

  4, 12, 36

Your fourth term is going to be equal to the third term (36) 
multiplied by the common ratio (3); this equals 108. So your fourth 
term is 108, and the first four terms are:

  4, 12, 36, 108

Notice how geometric sequences grow very fast!

I hope this helps. If you have other questions or you'd like to talk 
about this some more, please write back.

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