David has a Master of Business Administration, a BS in Marketing, and a BA in History.
After completing this lesson, you will be able to identify a finite sequence and different types of finite sequences. You will also be able to distinguish mathematical patterns of finite sequences.
What Is a Finite Sequence?
If something is finite, then it has a limit or is bounded. Suppose you wanted to list the colors of the rainbow. It is generally acknowledged that there are seven colors of the rainbow. You could show the colors in set notation as follows:
red, orange, yellow, green, blue, indigo, violet
The Colors of the Rainbow
The list of colors stops at seven, so the colors of the rainbow are a finite set.
In mathematics, a sequence is usually meant to be a progression of numbers with a clear starting point. Some sequences also stop at a certain number. In other words, they have a first term and a last term, and all the terms follow a specific order. This type of sequence is called a finite sequence. Let's go back to the rainbow example. Notice that the colors are listed as they appear in the rainbow, from the topmost down.
Now, let's look at a mathematical example. The first five positive odd numbers are an example of a finite sequence; it stops at the number 9:
1, 3, 5, 7, 9
The elements of a sequence are not an arbitrary list of numbers. In other words, they are not listed randomly, but follow a specific order. Often, finite sequences follow a specific mathematical pattern that can be represented by a general rule that can be displayed in algebraic terms or in words.
In a finite sequence, there is a first term, second term, and so on until the last term. The letter n often represents the total number of terms in a finite sequence. The first term of a finite sequence can be represented by a(1), the second term by a(2), and so on. The number in parentheses next to the a is usually written as a subscript, but parentheses will be used at times in this lesson.
This figure illustrates this nomenclature.
There are many different types of finite sequences, but we will stay within the realm of mathematics. An example of a finite sequence is the prime numbers less than 40 as shown below:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37
Another example is the natural numbers less than and equal to 100. It would be cumbersome to write out all of the terms in this finite sequence, so we can show it as follows:
1, 2, 3, 4, 5, …, 100
The ellipsis mark tells us that the pattern exhibited in the first five terms shown will continue until the last term of 100. There could be other sequences that start off the same way and end in 100 but that are not the natural numbers less than and equal to 100. In that case, it would be prudent to write the sequence sufficiently enough so that the reader understands the pattern.
Let's look at some finite sequences and identify any patterns.
A sequence in which all pairs of successive terms have a common difference is called an arithmetic finite sequence. Find the common difference in the following arithmetic finite sequence:
2, 7, 12, 17, …, 47
The first four terms of the sequence show that the common difference is 5. In other words, we can add 5 to any term in the sequence to get the next term in the sequence.
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A sequence in which all pairs of successive terms form a common ratio is called a geometric finite sequence. Find the common ratio in the following geometric finite sequence:
7, 21, 63, 189, 567
The sequence has a common ratio of 3 because 21/7 = 63/21 = 189/63 = 567/189 = 3. In other words, we can multiply any term in the sequence by 3 to get the next term in the sequence.
Describe the following finite sequence:
1/9, 2/8, 3/7, 4/6, …, 9/1
It is clear that this finite sequence is neither arithmetic nor geometric. But there certainly is a pattern that the sequence follows. Notice that the numerator of each term is the order of the term. For instance, a(1) has a numerator of 1, a(2) has a numerator of 2, and so on. What about the denominator? The denominator starts at 9 for the first term and decreases by 1 from one term to the next. We could say that the denominator is 10 less the order of the term. Let's establish an index for this sequence: a(i) will represent the ith term of the sequence. We can therefore write a general rule for this sequence as follows:
a(i) = i/(10 - i)
Let's try this with a few numbers in the sequence.
a(1) = 1/(10 - 1) = 1/9
a(2) = 2/(10 - 2) = 2/8
a(3) = 3/(10 - 3) = 3/7
An example of a sequence that is not finite would be one that is infinite. In other words, a sequence that has no end is an infinite sequence. The multiples of the number 5 would not be a finite sequence, because the list would be endless. We can always add 5 to the last multiple of 5 shown in the sequence to get a new multiple of 5.
The list of negative integers is another example of an infinite sequence and is shown in symbols below:
-1, -2, -3, -4, -5, …
The ellipsis symbol tells us that the sequence continues with no end.
Let's review. A finite sequence is a list of terms in a specific order. The sequence has a first term and a last term. The order of the terms of a finite sequence follows some type of mathematical pattern or logical arrangement. If an ellipsis mark is used to display a finite sequence, it is important that the pattern or arrangement of the sequence is not ambiguous and clearly defined to the reader.
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