The concept of a sequence is not difficult to understand; a sequence is simply an ordered list of numbers. We call the numbers in a sequence terms

For example, the following is a sequence:

1, 2, 3, 4,...

The three dots mean that it continues on forever. Because it continues forever, we refer to it as aninfinite sequence. Here is an example of a finite sequence:

5, 10, 15, 20

This sequence has only four terms.

Strictly speaking, any set of ordered numbers is a sequence - even numbers which don't appear to have any particular pattern:

7, 0, -1, 10, -8, 0...

However, these kinds of sequences are not usually as interesting. The kinds of sequences which are most interesting to mathematicians are Arithmetic Sequences and Geometric Sequences

Arithmetic Sequences
An arithmetic sequence is a sequence in which every pair of numbers has a common difference. For example, the following is an arithmetic sequence because the difference between every pair of terms is 4:

5, 9 , 13, 17,...

Since we know that the common difference is 4, we can find the next term by adding 4 to 17, which gives us 21.

The common difference can be negative:

11, 9, 7, 5,...

In this case the common difference is -2, and the next term is 3.

Geometric Sequences
A geometric sequence is a sequence in which every pair of numbers has a common ratio. For example the following is a geometric sequence because the ratio between every pair of terms is 2:

1, 2, 4, 8, ...

As you can see, the terms get large very quickly. The next term is 16, because 8 x 2 = 16.

Geometric sequences can have ratios that are fractions, like this one, which has a common ratio of 1/2:

24, 12, 6, 3,...

Every term is found by multiplying the previous term by 1/2, so the next term is 3/2 or 1.5.

What happens if a geometric sequence has a negative ratio? Well, let's see:

3, -6, 12, -24, 48,...

You can see that the terms alternate between positive and negative. The ratio here is -6/3 = -2. So the next term will be 48 x (-2) = -96.

In each of the problems below, state if the sequence is arithmetic or geometric, and find the next term.