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A sequence is an ordered list. It is a function whose domain is the natural numbers {1, 2, 3, 4, ...}. |
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Sequence: |
1, |
5, |
9, |
13, |
17, |
21, |
... |
Notation for terms
of the sequence: |
a1 |
a2 |
a3 |
a4 |
a5 |
a6 |
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 Information about sequences:
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Each number in a sequence is called a term, an element or a member. |
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Terms are referenced in a subscripted form (indexed), where the natural number subscripts (often referred to by n) are {1, 2, 3, ...}and refer to the location (position) of the term in the sequence. The first term is denoted a1, the second term a2, and so on. The nth term is an.
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The terms in a sequence may, or may not, have a pattern or related formula.
Example: the digits of π form a sequence, but do not have a pattern. |
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A subscripted form of a sequence is represented by a1, a2, a3, ... an, ... |
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A functional form of a sequence is represented by f (1), f (2), f (3), ..., f (n),... |
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Sequences are functions. |
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The domain of a sequence consists of the natural (counting) numbers 1, 2, 3, 4, ... |
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The range of a sequence consists of the terms of the sequence. |
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When graphed, a sequence is a scatter plot, a series of dots. (Do not connect the dots). |
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The sum of the terms of a sequence is called a series.
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Sequences can be expressed in various forms:
Term Number
|
Term |
Subscript Notation |
Function Notation |
1 |
1 |
a1 |
f (1) |
2 |
5 |
a2 |
f (2) |
3 |
9 |
a3 |
f (3) |
4 |
13 |
a4 |
f (4) |
5 |
17 |
a5 |
f (5) |
6 |
21 |
a6 |
f (6) |
n |
|
an |
f (n) |
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{1, 5, 9, 13, 17, 21, ...} (list)
Subscripted notation:
an= 4n - 3 (explicit form)
a1 = 1; an= an-1 + 4 (recursive form)
Functional notation:
f (n) = 4n - 3 (explicit form)
f (1) = 1; f (n) = f (n - 1) + 4 (recursive form)
Note: Not all functions can be defined by an explicit and/or recursive formula.
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Forms of sequences:
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A finite sequence contains a finite number of terms (a limited number of terms) which can be counted.
Example: {1, 5, 9, 13, 17} (it starts and it stops) |
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An infinite sequence contains an infinite number of terms (terms continue without end) which cannot be counted.
Example: {1, 5, 9, 13, 17, 21, ...} (it starts but it does not stop, as indicated by the ellipsis ... ) |
Ways of expressing (defining) sequences:
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A sequence may appear as a list (finite or infinite):
Examples: {1, 5, 9, 13, 17} and {1, 5, 9, 13, 17, 21, ...}
Listing makes it easy to see any pattern in the sequence. It will be the only option should the sequence have no pattern. |
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A sequence may appear as an explicit formula. An explicit formula designates the nth term of the sequence, an , as an expression of n (where n = the term's location).
Example: {1, 5, 9, 13, 17, 21, ...} can be written an = 4n - 3.
(a formula in terms of n) Read more at Sequences as Functions - Explicit |
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A sequence may appear as a recursive formula. A recursive formula designates the starting term, a1, and the nth term of the sequence, an , as an expression containing the previous term (the term before it), an-1.
Example: {1, 5, 9, 13, 17, 21, ...} can be written a1 = 1; an= an-1 + 4.
(two-part formula in terms of the preceding term) Read more at Sequences as Functions - Recursive.
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Graphing Sequences:
Sequence: {1, 5, 9, 13, 17, 21, 25, 29, ...}
• Sequences are functions. They pass the vertical line test for functions.
• The domain consists of the natural numbers, {1,2,3,...}, and the range consists of the terms of the sequence.
• The graph will be in the first quadrant and/or the fourth quadrant (if sequence terms are negative).
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• Arithmetic sequences are linear functions. While the n-value increases by a constant value of one, the f (n) value increases by a constant value of d, the common difference. The rate of change is a constant "d over 1", or just d.
• Geometric sequences are exponential functions. While the n-value increases by a constant value of one, the f (n) value increases by multiples of r, the common ratio. The rate of change is not constant, but increases or decreases over the domain. |
Popular sequence patterns:
You should always be on the lookout for patterns, such as those shown below, when working with sequences. Keep in mind, however, that while all sequences have an order, they may not necessarily have a pattern.
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Arithmetic Sequence: (where you add (or subtract) the same value to get from one term to the next.) If a sequence adds a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The number added to each term is constant (always the same) and is called the common difference, d. The scatter plot of this sequence will be a linear function. |
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Geometric Sequence: (where you multiply (or divide) the same value to get from one term to the next.) If a sequence multiplies a fixed amount from one term to the next, it is referred to as a geometric sequence. The number multiplied is constant (always the same) and is called the common ratio, r. The scatter plot of this sequence will be an exponential function. |
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Doubting Thomas wonders how we can know, for sure, that a sequence such as 2, 4, 6, 8, ... is an arithmetic sequence. His theory is that there could be many other possible patterns, such as: 2, 4, 6, 8, 2, 4, 6, 8, ... (repeating 4 terms is his pattern).
Yes, Thomas is correct. Without a specification in the problem, there is the possibility of more than one pattern in most sequences. The person creating the sequence may have been thinking of a different pattern than what you see when you look at the sequence. In Algebra 1, if in doubt, first look for arithmetic or geometric possibilities. |
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Note: The indexing (subscripts) used for sequences can begin with 0 or any positive integer. The most popular indexing, however, begins with 1 so the index can also represent the position of the term in the sequence. Unless otherwise stated, this site will start indexes at 1. |
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Note: Computer programming languages such as C, C++ and Java, refer to the starting position in an array with a subscript of zero. Programmers must remember that a subscript of 3 refers to the 4th element, not the 3rd element, in the array. |
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