
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 nth 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.
Terms
Formulae
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}(). 