Math 12 – Summary of
Sequences
A sequence is an ordered list of terms that is created according to some pattern or rule. A sequence is written inside of curly brackets ( i.e. "{" and "}"). If a sequence ends with an ellipsis (...) then it continues on with the same pattern and is called an infinite sequence. If there is no ellipsis then it is a finite sequence.
An arithmetic sequence is a sequence in which each consecutive term changes by the same amount. The amount that the terms in an arithmetic sequence change is called the common difference.
Example: The sequence {2, 5, 8, 11, 14, 17, ... } is an arithmetic sequence with an infinite number of
terms. The first term,
, is 2 and the common difference, d, is 3. The rule for
generating the terms of this sequence is 
.
A power sequence is any sequence that has a sequence of differences at any level except the first. Quadratic, cubic and quartic sequences are all power sequences.
A geometric sequence is a sequence in which each consecutive term is multiplied by the same amount. The amount that the terms in a geometric sequence are multiplied by is called the common ratio.
Example: The sequence {3, 6, 12, 24, 48, ... } is a geometric sequence with an infinite number of
terms. The first term,
, is 3 and the common ratio, r, is 2. The rule for
generating the terms of this sequence is 
.
Summary of Sequences
|
Name |
How do I recognize it? |
Degree |
General Form |
|
Arithmetic |
The sequence of first-level differences,
|
1 |
|
|
Quadratic |
The sequence of second-level differences, |
2 |
|
|
Cubic |
The sequence of third-level differences, |
3 |
|
|
Quartic |
The sequence of fourth-level differences, |
4 |
|
|
Geometric |
There is a common ratio, r |
n |
|
, are all the same.
, are all the same
, are all the same
, are all the same
