Arithmetic and Geometric Sequences

Vocabulary terms and definitions regarding arithmetic and geometric sequences, their common properties, and their graphical representations.

Arithmetic Sequence

A sequence of numbers in which the difference between any two consecutive terms is a constant, meaning a constant is added to each term to produce the next term.

Common Difference

The positive or negative constant added to each term in an arithmetic sequence to determine the next value, represented by the variable $d$.

Geometric Sequence

A sequence of numbers in which the ratio between any two consecutive terms is a constant, meaning each term is multiplied by a constant to determine the next term.

Common Ratio

The integer or fraction constant by which each term in a geometric sequence is multiplied, represented by the variable $r$.

$$a_n$$

Notation representing the $n^{th}$ term of a sequence.

$$a_{n-1}$$

Notation representing the term immediately before the $n^{th}$ term ($a_n$).

Consecutive Terms

Terms that follow each other in order within a mathematical sequence.

Sequence Graph

A set of discrete points representing the values of a sequence.

Arithmetic Sequence Graph

A graph where the points lie on a line; it is increasing when the common difference is positive and decreasing when the common difference is negative.

Geometric Sequence Graph ($r > 1$)

A graph where the points do not lie on a line and the function is increasing.

Geometric Sequence Graph ($0 < r < 1$)

A graph where the points do not lie on a line and the function is decreasing.

Geometric Sequence Graph ($r < 0$)

A graph where the points do not lie on a line and the values alternate between increasing and decreasing between consecutive points.