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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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Terms are referenced in a subscripted form (indexed), where the natural number subscripts, 1, 2, 3, ..., 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.
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.
Sequences can be expressed in various forms:
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.
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.
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.
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.
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.
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.