Sequence
A collection of numbers in a set order. {1, 4, 7, 10, …} is an example of a sequence for which the first four terms are specified (but the sequence continues beyond these four terms, as indicated by the “…”)
Nth Term
A particular term in a sequence. The term number, n, corresponds to the term’s location in a sequence. In the sequence {1, 4, 7, 10, …}, 1 is the 1st term (n = 1), 4 is the 2nd term (n = 2), 7 is the 3rd term (n = 3), and so on.
Value of a Sequence Term
The value of a particular term in the sequence. In the sequence {1, 4, 7, 10, …} the 1st term has a value of 1, the 2nd term has a value of 4, the 3rd term has a value of 7, and so on.
Sequence Rule
The rules that determine the order of numbers in a given sequence. In the sequence {1, 4, 7, 10, …}, each term is 3 more than the previous term, so the rule is to add 3 each time to get the next term. Rules can also be written as direct or recursive sequence formulas (see below).
Direct Sequence Formula
One way to write a sequence formula. A direct sequence is defined as a function of n, the place in which the term occurs in the sequence. For the sequence {1, 4, 7, 10, …}, the direct sequence formula is An = 3n – 2, for integers n >= 1.
Recursive Sequence Formula
Another way to write a sequence formula. A recursive sequence is defined in terms of the value of previous items in the sequence. For the sequence (1, 4, 7, 10, …}, the recursive sequence formula is An = An-1 + 3.
Linear Sequence
A sequence in which the difference between successive terms is always the same. A constant number (which could be negative!) is added each time. Also called Arithmetic Sequence.
Direct Linear (or Arithmetic) Sequence
One way to write the direct linear sequence formula is Sn = kn + x where k is the constant difference between successive terms, x is some other constant, and n is the number of the term in question. Another way to write the direct linear sequence formula is Sn = S1 + (n - 1)k, where S1 is the value of the first term in the sequence, n is the number of the term in question, and k is the constant difference between successive terms.
Recursive Linear (or Arithmetic) Sequence
The recursive linear sequence formula is Sn = Sn-1 + k, where Sn-1 is the value of the previous term in the sequence and k is the constant difference between successive terms. In addition to the recursive formula, the value of one specific term must be given, along with its term number. For example, S2 = 6 tells us that the 2nd term of the sequence has the value 6.
Arithmetic Sequence
See Linear Sequence (above).
Exponential Sequence
A sequence in which the ratio between successive terms is always the same; a constant number (which could be negative!) is multiplied each time.
Direct Exponential (or Geometric)Sequence
The standard formula is Sn = x(kn), where x is the value of the first term in the sequence, k is the value of the ratio (the number by which we multiply each successive term), and n is the number of the term in question.
Recursive Exponential (or Geometric) Sequence
The standard formula is Sn = kSn-1, where k is the value of the ratio (the number by which we multiply each successive term), and n is the number of the term in question. In addition to the recursive formula, the value of one specific term must be given, along with its term number. For example, S2 = 6 tells us that the 2nd term of the sequence has the value 6.
Geometric Sequence
See Exponential Sequence (above).
