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C4 - Sequences

Page history last edited by PBworks 13 years, 7 months ago

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 Formula

1

 Formula

  or Formula

 {3, 5, 7, 9, ...}

Formula

Quadratic  There is a common difference at Formula

2

 Formula

 {2, 7, 16, 29, 46 ...}

tn = 2n2 - n + 1

Cubic  There is a common difference at Formula

3

 Formula

 {1, 8, 27, 64, 125, ...}

tn = n3

Quartic

 There is a common difference at Formula

4

 Formula

 {2, -11, -22, 5, 130, 437, ...}

tn = n4 - 4n3 + 5

Quintic  There is a common difference at Formula

5

 Formula

 {2, 33, 244, 1025, 3126, ...}

tn = n5 + 1 

Geometric  There is a common ratio

n

 Formula

 {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.

 

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