Terms
Arithmetic Sequence

A sequence in which each term is a constant amount greater or less than
the previous term. In this type of sequence,
a
_{n+1} = a
_{n} + d
, where
d
is
a constant.
Common Ratio

In a geometric sequence, the ratio
r
between each term and the
previous term.
Convergent Series

A series whose limit as
n→∞
is a real number.
Divergent Series

A series whose limit as
n→∞
is either
∞
or
 ∞
.
Explicit Formula

A formula for the
n
th term of a sequence of the form
a
_{n} =
some function
of
n
.
Finite Sequence

A sequence which is defined only for positive integers less than or equal to
a certain given integer.
Finite Series

A series which is defined only for positive integers less than or equal to a
certain given integer.
Geometric Sequence

A sequence in which the ratio between each term and the previous term is
a constant ratio.
Index of Summation

The variable in the subscript of
Σ
. For
a
_{n}
,
i
is the
index of summation.
Infinite Sequence

A sequence which is defined for all positive integers.
Infinite Series

A series which is defined for all positive integers.
Recursive Sequence

A sequence in which a general term is defined as a function of one or
more of the preceding terms. A sequence is typically defined recursively by
giving the first term, and the formula for any term
a
_{n+1}
after the first
term.
Sequence

A function which is defined for the positive integers.
Series

A sequence in which the terms are summed, not just listed.
Summation Notation

a
_{n} = a
_{1} + a
_{2} + a
_{3} + a
_{4} + ... + a
_{n}
. The symbol
Σ
and
its subscript and superscript are the components of summation notation.
Term

An element in the range of a sequence. A sequence is rarely represented by
ordered pairs, but instead by a list of its terms.
Limit of an Infinite Geometric Series

For a geometric sequence
a
_{n} = a
_{1}
r
^{n1}
, where
1 < r < 1
, the limit
of the infinite geometric series
a
_{1}
r
^{n1} =
. This is the same as the sum of the infinite geometric
sequence
a
_{n} = a
_{1}
r
^{n1}
.

Sum of a Finite Arithmetic Sequence

The sum of the first
n
terms of the arithmetic sequence is
S
_{n} = n()
or
S
_{n} = na
_{1} + (dn  d )
, where
d
is the
difference between each term.

Sum of a Finite Geometric Sequence

For a geometric sequence
a
_{n} = a
_{1}
r
^{n1}
, the sum of the first
n
terms is
S
_{n} = a
_{1}()
.
