IndietroSequences: Types, Properties, and Representations
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Sequences, Series, & Induction
Types of Sequences
Sequences are ordered lists of numbers, where each term follows a specific pattern or rule. Two fundamental types of sequences in College Algebra are arithmetic sequences and geometric sequences.
Arithmetic Sequence: Each term is obtained by adding a constant value, called the common difference, to the previous term.
Geometric Sequence: Each term is obtained by multiplying the previous term by a constant value, called the common ratio or growth factor.
Definitions
Common Difference (Arithmetic): The fixed amount added to each term. It can be found by subtracting consecutive terms:
Common Ratio (Geometric): The fixed amount multiplied to each term. It can be found by dividing consecutive terms:
Examples
Arithmetic Sequence Example: 3, 5, 7, 9, 11, ... Common difference:
Geometric Sequence Example: 400, 20, 4, ... Common ratio:
Representing Sequences
Sequences can be represented in several ways, including as a list, a table, a graph, or using function notation and equations.
List: Simply writing out the terms in order.
Table: Organizing terms with their position (n).
Graph: Plotting the terms against their position.
Function Notation: Using to denote the nth term.
Equation: Expressing the rule for generating terms.
Recursive and Explicit Definitions
Recursive Definition: Defines each term based on the previous term(s) and a rule. Example (arithmetic): ,
Explicit Definition: Defines each term directly in terms of its position. Example (arithmetic):
Key Properties
Domain: The set of positions (usually positive integers) for which the sequence is defined.
First Term: The starting value of the sequence, often denoted .
Rule: The formula or pattern used to generate the sequence.
Summary Table: Arithmetic vs. Geometric Sequences
Type | Rule | Common Value | Example |
|---|---|---|---|
Arithmetic | Common Difference () | 3, 5, 7, 9, ... | |
Geometric | Common Ratio () | 400, 20, 4, ... |
Example Application: To find the 5th term of an arithmetic sequence with and :
Example Application: To find the 4th term of a geometric sequence with and :
Additional info: The notes reference "induction" in the chapter title, but do not provide details. Mathematical induction is a method of proof used to establish properties of sequences, but is not covered in the provided content.
