Where the terms increase/decrease by the addition/subtraction of a fixed amount. The difference of two terms next to each other is called the common difference.
Example:
So the common difference would be 3.
You could write that algebraically as either:
ak = 2 + 3k for k = 1, 2, 3, ...
When k = 1, a1 = 2 + 3 = 5
k = 2, a2 = 2 + 6 = 8
k = 3, a3 = 2 + 9 = 11
etc.
or
a1 = 5
ak+1 = ak + 3 for k = 1, 2, 3, ...
Substituting k = 1, a2 = a1 + 3 = 5 + 3 = 8
k = 2, a3 = a2 + 3 = 8 + 3 = 11
k = 3, a4 = a3 + 3 = 11 + 3 = 14
etc.
The first example defines the value of ak and is called the deductive definition. The second defines each term by looking at the previous and is so called inductive so you need to know both for different situations.
Geometric Sequences
Each term is the previous term multiplied by a fixed number - called the common ratio.
Example :
This example has a common ratio of 2.
Algebraically, it's written as:
ak = 5 × 2k for k = 1, 2, 3...
so a1 = 10
or
ak + 1 = 2ak for k = 1, 2, 3...
The first example is called deductive definition and the second is called inductive definition.
Periodic Sequences
Repeats itself in regular intervals. An example could be the number of hours a restaurant is open each day:
a1 = 8, a2 = 0, a3 = 10, a4 = 10, a5 = 10, a6 = 10, a7 = 12, a8 = 8, a9 = 0...
(Sun) (Mon) (Tues) (Wed) (Thurs) (Fri) (Sat) (Sun) (Mon)
In general, you can see that a8 = a1 and a9 = a2 etc.
Also, ak + 7 = ak for k = 1, 2, 3... as the sequence has a period of 7 (each term is repeated after seven terms).
So if ak + p = ak for k = 1, 2, 3... (for a fixed integer, p), the sequence is periodic, the period being the smallest positive value of p.
Oscillating Sequences
Go above and below a certain middle number.
Example: 5, 6, 5, 4, 5, 6, 5, 4
The middle number being 5.
That example is also periodic with period of 4, as after 4 numbers the sequence repeats itself.
Sequences With Other Patterns
There are so many patterns that can be made with sequences as well as the ones we've looked at above. However, if you subtract each term from the next, you could find that the differences form a pattern.
Example: 1, 4, 9, 16, 25 has the differences of 3, 5, 7 and 9.
The differences in this example make an arithmetic sequence with the common difference of 2. The sequence could also be written as: 12, 22, 32, 42, 52...