Outcome: C4 demonstrate an understanding of patterns that are arithmetic, power, and geometric and relate them to corresponding functions
Originator: Mr. Lee
Explanation: A sequence is an ordered list of terms that is generated by 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 sequences is a specific type of sequence that has a common difference. Aritmetic sequences relate to linear functions, power sequences relate to polynomial functions with a degree greater than one, and geometric sequences relate to exponential functions.
Summary of Sequences
Name  How do I recognize it?  Degree  General Form  Example 
Arithmetic  There is a common difference at 
1 
or 
{3, 5, 7, 9, ...} 
Quadratic  There is a common difference at 
2 
{2, 7, 16, 29, 46 ...} t_{n} = 2n^{2}  n + 1 

Cubic  There is a common difference at 
3 
{1, 8, 27, 64, 125, ...} t_{n} = n^{3} 

Quartic 
There is a common difference at 
4 
{2, 11, 22, 5, 130, 437, ...} t_{n} = n^{4}  4n^{3} + 5 

Quintic  There is a common difference at 
5 
{2, 33, 244, 1025, 3126, ...} t_{n} = n^{5} + 1 

Geometric  There is a common ratio 
n 
{3, 6, 12, 24, 48, ...} t_{n} = 3(2)^{(n1)} 
Sample Problem:
Determine the type of the following sequence: {0, 10, 12, 0, 32, 90, ...}
Solution:
Step 1: Check to see if the sequence has a common ratio.
10/0 = DNE, 12/10 = 1.2, 0/12 = 0
There is no common ratio so this is NOT a geometric sequence
Step 2: Determine the Sequences of Differences until a common difference is found
D1 = {10, 2, 12, 32, 58, ...}
D2 = {8, 14, 20, 26, ...}
D3 = {6, 6, 6, ...}
There is a common difference on the third level of differences therefore this is a CUBIC sequence.
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