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Finite Sequences and Series
Mathematicians define a sequence as a function which maps some subset of the positive integers onto some subset of the real numbers. In simplest terms, a sequence is an ordered list of elements. In this article, we will take these elements to be real numbers to simplify the discussion. A finite sequence, then, is a finite ordered list of real numbers: a1,a2,a3,...,an. Thus 1,2,4,8,16,32 is a finite sequence, as is -1,0,1,-1,0,1, and 1,3,5,7,9,...,99. When a finite sequence has more terms than we care to write down, we use three dots (...) to designate the numbers we are not listing individually. The three sequences given above all have a clearly discernable pattern, but that is not a requirement for sequences in general. It does happen that most of the sequences that are important in mathematics do have some kind of pattern that allows one to find a general formula for the nth term, or, at least, a recursive formula that allows one to link each term to the one preceding it in the sequence. For example, in the sequence 1,2,4,8,16,32, we can see that each term is a power of 2. The first term is 20 = 1; the second term is 21 = 2; the third term is 22; and so on. Therefore we can say that, in general, the nth term of this sequence is given by 2n-1. We can also describe this sequence by a recursive rule that relates each term of the sequence to the one before it. Since we obtain each term after the first by multiplying the preceding term by 2, our recursive definition could be written as a1 = 1 and ak+1 = 2ak. Note that when writing the recursive rule we must give the value of the first term and the relationship of each term to the one preceding it in the sequence. For the sequence 1,3,5,7,9,...,99, each term is an odd number that is 2 greater than the term before it. The general term can be written as an = 2n - 1. A recursive rule would be a1 = 1 and ak+1 = ak + 2.
The idea is that if you know the first term of a sequence and the rule which tells you how to get any term from the one preceding it, then you know everything about the sequence. In principle, you could write down the first term, use it to get the second term, use the second term to get the third, and so on. A recursive rule is very effective in programming a computer or calculator to generate hundreds or thousands of terms of a sequence, as is sometimes done in computer simulations of some natural processes in which each step in the process depends on the preceding step.
Two of the most important types of sequences in mathematics are geometric sequences and arithmetic sequences. We have seen an example of each kind in the preceding paragraph. The sequence 1,2,4,8,16,32 is geometric, also called exponential because of its relationship to exponential functions. In fact, each member of this sequence is a value of the exponential function f(x) = 2x. In general, a finite geometric sequence looks like a,ar,ar2, ar3,..., arn-1. Using this notation, a is the first term of the geometric sequence and r is called the "common ratio." Essentially, one obtains each term of this sequence by multiplying the preceding term by r, hence the name common ratio. In 1,2,4,8,16,32, the first term is 1 and the common ratio is 2. The sequence 1,3,5,7,9,...,99 is arithmetic. A general notation for finite arithmetic sequences is a, a+d, a+2d, a+3d,..., a+(n-1)d. Again a is the first term, as in the geometric sequence, but the d is called the "common difference." Here one obtains a term of the sequence by adding d to term before it. In the sequence 1,3,5,7,9,..., 99, the first term is 1 and the common difference is 2.
Once we have a basic understanding of finite sequences, we can give the definition of a finite series: A finite series is the sum of the terms of a finite sequence. So, whereas, 1,2,4,8,16,32 is a finite sequence, 1 + 2 + 4 + 8 + 16 + 32 is a finite series. In fact, it is called a finite geometric series because it is the sum of the terms of a finite geometric sequence. Likewise, 1 + 3 + 5 + 7 + 9 + ... + 99 is a finite arithmetic series. Thus, given any finite sequence, we can create a corresponding finite series by simply adding together all the terms of the sequence. An understanding of finite sequences and series forms the necessary background for the study of infinite sequences and series, which are studied extensively in calculus and other advanced mathematics courses.
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