Terms and Formulae
Terms
Arithmetic Sequence
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A sequence in which each term is a constant amount greater or less than
the previous term. In this type of sequence, an+1 = an + d, where d is
a constant.
Common Ratio
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In a geometric sequence, the ratio r between each term and the
previous term.
Convergent Series
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A series whose limit as n→∞ is a real number.
Divergent Series
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A series whose limit as n→∞ is either ∞ or - ∞.
Explicit Formula
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A formula for the nth term of a sequence of the form an = some function
of n.
Finite Sequence
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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
an,
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 an+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
-
an = a1 + a2 + a3 + a4 + ... + an. 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.
Formulae
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Limit of an Infinite Geometric Series
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For a geometric sequence an = a1rn-1, where -1 < r < 1, the limit
of the infinite geometric series  a1rn-1 =  . This is the same as the sum of the infinite geometric
sequence an = a1rn-1.
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Sum of a Finite Arithmetic Sequence
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The sum of the first n terms of the arithmetic sequence is Sn = n( ) or Sn = na1 +  (dn - d ), where d is the
difference between each term.
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Sum of a Finite Geometric Sequence
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For a geometric sequence an = a1rn-1, the sum of the first n
terms is Sn = a1( ).
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