Sequences: Types, Properties, and Representations

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: $d = a_{n} - a_{n-1}$

• Common Ratio (Geometric): The fixed amount multiplied to each term. It can be found by dividing consecutive terms: $r = \frac{a_{n}}{a_{n-1}}$

Examples

• Arithmetic Sequence Example: 3, 5, 7, 9, 11, ... Common difference: $d = 5 - 3 = 2$

• Geometric Sequence Example: 400, 20, 4, ... Common ratio: $r = \frac{20}{400} = 0.05$

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 $a_n$ 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): $a_1 = 3$, $a_{n} = a_{n-1} + 2$

• Explicit Definition: Defines each term directly in terms of its position. Example (arithmetic): $a_n = 3 + 2(n-1)$

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 $a_1$.

• Rule: The formula or pattern used to generate the sequence.

Summary Table: Arithmetic vs. Geometric Sequences

Type	Rule	Common Value	Example
Arithmetic	$a_n = a_1 + (n-1)d$	Common Difference ($d$)	3, 5, 7, 9, ...
Geometric	$a_n = a_1 \cdot r^{n-1}$	Common Ratio ($r$)	400, 20, 4, ...

Example Application: To find the 5th term of an arithmetic sequence with $a_1 = 3$ and $d = 2$: $a_5 = 3 + (5-1) \times 2 = 3 + 8 = 11$

Example Application: To find the 4th term of a geometric sequence with $a_1 = 400$ and $r = 0.05$: $a_4 = 400 \times (0.05)^{3} = 400 \times 0.000125 = 0.05$

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.