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 ...} tn = 2n2 - n + 1
Cubic	There is a common difference at	3		{1, 8, 27, 64, 125, ...} tn = n3
Quartic	There is a common difference at	4		{2, -11, -22, 5, 130, 437, ...} tn = n4 - 4n3 + 5
Quintic	There is a common difference at	5		{2, 33, 244, 1025, 3126, ...} tn = n5 + 1
Geometric	There is a common ratio	n		{3, 6, 12, 24, 48, ...} tn = 3(2)(n-1)

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.

Links: 

• Purplemath.com - Arithmetic and Geometric Sequences