Arithmetic & Geometric Sequences
VOCABULARY:
sequence, term(s) of a sequence, arithmetic sequence, common difference, geometric sequence, ratio, recursive formula, explicit formula
sequence: ordered list of #’s that often form a pattern. It’s a function whose domain is the set of natural numbers.
terms of a sequence: each # in the list, also known as the output, or range, of the sequence
NOTES: For either type of sequence (arithmetic or geometric) and their formulas (recursive and explicit), it is important to know the first term and
the special relationship, or pattern, between the terms.
When given a sequence, determine if there is a pattern between the terms.
First, try to determine if there is a common difference by subtracting the preceding term from any subsequent term, after the first term. If
there is a constant difference, then it represents an arithmetic sequence.
If there is no common difference, check to see if a common ratio can be found between any terms and their preceding terms. If there is a
constant ratio, then it represents a geometric sequence.
Sequences can be written using 2 formulas: recursive & explicit
TYPE OF
SEQUENCE
Arithmetic
Sequence
DEFINITION
A sequence where the difference
between consecutive terms is a
constant value.
common difference “d”: the
constant difference between
consecutive terms in an arithmetic
sequence
Geometric
Sequence
A sequence where the ratio, “r,” of
any term to its preceding term is a
constant value.
RECURSIVE FORMULA
(Useful for finding the next term)


Uses addition (+)
A function rule that relates each term of
a sequence after the first term to the
ones before it.

A(n) = A(n-1) + d, where A(1) =_.

(List the first term after the formula.)
To find the next term, you need to know
the one before it.


Uses multiplication (x)
a1 = a

an = an-1 ∙ r, n ≥ 2
EXPLICIT FORMULA
(Useful for finding the nth term)




Uses multiplication (x)
Linear Functions are the explicit form of
recursively-defined arithmetic sequences.
A function rule that relates each term of a
sequence to the term number.
This gives an ordered pair for graphing.
(term # or position is the x-value, term of
the sequence or y-value)
 A(n) = A(1) + (n-1)d




Uses exponentiation (rn-1)
Exponential Functions are the explicit
form of recursively-defined geometric
sequences.
an = a1 ∙ rn-1, n ≥ 1
an can be written as f(x)