Lesson Explainer: Introduction to Sequences Mathematics

In this explainer, we will learn how to identify a sequence and some of the common properties of sequences.

A fundamental notion in mathematics is that of a “sequence,” which is generally taken to represent an ordered list of numbers. Common sequences would include the integers 1,2,3,4,, the square numbers 1,4,9,16,, other recognizable sequences such as 2,4,8,16,, and other related sequences. These are examples of sequences where there are infinitely many elements, although we could have taken any finite subset of these and this would still have represented a sequence. It is important to note that the sequence 1,2,3,4 is not the same as the sequence 1,2,4,3 and in this sense the notion of a sequence is distinct from that of a set, where the ordering of the objects does not matter. One further distinction is that, unlike a set, a mathematical sequence may have repeated elements.

The notion of a sequence is so wide-ranging that it would be of little worth to try and give a proper introduction within the confines of this explainer. Instead, we will restrict our gaze to focusing only on two types of common sequences that appear throughout mathematics and many related areas: arithmetic sequences and geometric sequences. These are both examples of sequences where each term is generated by some combination of the previous term or terms which are often referred to as “iterative” sequences or “recursive” sequences. These types of sequences have a significant number of features and applications that are born from strong links to the most fundamental operations in mathematics, as we will describe throughout the explainer. We will begin with arithmetic sequences before progressing onto geometric sequences later in the explainer.

Definition: Arithmetic Sequence

An “arithmetic” sequence is one in which each term can be obtained from the previous term by means of adding a common difference 𝑑. In other words, the sequence 𝑇 is arithmetic if 𝑇=𝑇+𝑑 for any value of 𝑛 where 𝑛 is taken to be a natural number. It can often be more useful to rephrase this requirement as 𝑇𝑇=𝑑 for all values of 𝑛. By shifting 𝑛𝑛1, we can alternatively express these conditions as 𝑇=𝑇+𝑑 or as 𝑇𝑇=𝑑, with the latter being used as the standard representation in some of the literature.

For a sequence to be arithmetic, the above property must apply to every single term. If, at some point in the sequence, there is an exception to this property, then the sequence is not arithmetic. We will illustrate this with the following example. Consider the finite sequence 3,9,15,21. We can label the general term as 𝑇 and hence list each element of this sequence in turn, giving 𝑇=3,𝑇=9,𝑇=15,𝑇=21.

We can now begin to apply this criteria to deduce whether or not this sequence is arithmetic. By using the property stated in the above definition, we calculate that 𝑇𝑇=93=6,𝑇𝑇=159=6,𝑇𝑇=2115=6.

We have therefore found that 𝑇𝑇=6 for all values of 𝑛, which means that this sequence is arithmetic.

We could easily destroy this property by adding a further term to the sequence, which violates the required conditions. For example, we could append the following value to the sequence: 𝑇=22, making the full statement of the sequence now 3,9,15,21,22. In this case, we would then have that 𝑇𝑇=2221=1.

Clearly, it is not the case that 𝑇𝑇=6 for all values of 𝑛 and hence this new sequence is not arithmetic.

Naturally, the definition of an arithmetic sequence allows for the case where the common difference 𝑑 is a negative number, in which case each term of the sequence would be less than the term that preceded it. This definition therefore incorporates all iterative sequences where the terms are related to each other directly by addition of a constant term, hence incorporating two of the most basic operations. Now, we will move onto a different type of sequence where each term can be obtained by multiplication of the previous term (which will implicitly include the notion of division).

Definition: Geometric Sequence

A “geometric” sequence is one in which each term can be obtained from the previous term by means of multiplying by a common ratio 𝑟. In other words, the sequence 𝑇 is geometric if, 𝑇=𝑇×𝑟 for any value of 𝑛, where 𝑛 is a natural number. This criteria can alternatively be expressed as 𝑇𝑇=𝑟 for all values of 𝑛. By shifting 𝑛𝑛1, we can alternatively express these conditions as 𝑇=𝑇×𝑟 or as 𝑇𝑇=𝑟, with the latter being used as the standard representation in some of the literature.

As with arithmetic sequences, we will illustrate the definition of a geometric sequence by way of example. Consider the finite sequence 1,2,4,8, which we recognize as the first four powers of 2 (including the zeroth power). We will label the general term as 𝑇, as above, and then list each element of this sequence in turn, giving 𝑇=1,𝑇=2,𝑇=4,𝑇=8.

Now, we begin investigating whether there is any common ratio 𝑟 that can be used to obtain each term from the previous term, according to the above definition of a geometric sequence. We compute 𝑇𝑇=21=2,𝑇𝑇=42=2,𝑇𝑇=84=2.

We observe that 𝑇𝑇=2 for all values of 𝑛, meaning that we are working with a geometric sequence with a common ratio 𝑟=2.

In order to now annihilate this property, we append a fifth value to this sequence, as follows: 𝑇=12.

The new sequence is 1,2,4,8,12. If we find the ratio of the fifth and the fourth terms, we find that 𝑇𝑇=128=32.

This is not equal to the common ratio 𝑟=2; hence, we have found one example where it is not the case that 𝑇𝑇=2 for all 𝑛. This implies that the new sequence is not geometric.

With arithmetic sequences, each term is either larger or smaller than the previous term (depending on whether the common difference term 𝑑 is positive or negative respectively). However, for geometric sequences this need not be the case. As we will see in later examples, a geometric sequence can alternate between positive and negative values if the value of the ratio is negative.

When trying to deduce whether or not a sequence is arithmetic or geometric, there are few tests more effective than the direct application of the two definitions that we gave above. With practice, it becomes fairly trivial to recognize common sequences and to classify them appropriately. If the matter is ever in doubt, then the definitions should be used, as we will show in the following example.

Example 1: Identifying a Sequence That Is Neither Arithmetic nor Geometric

Which of the following sequences is not be classified as arithmetic or geometric?

  1. 12,1,32,2,
  2. 12,13,14,15,
  3. 12,14,18,116,
  4. 13,13,1,53,
  5. 1,12,0,12

Answer

For each of the following, we will only give a brief description of our methods for determining whether or not the sequences are geometric or arithmetic. We will assume that the sequence continues infinitely in a way that respects the initial properties that are observed.

  1. We will label this sequence as 𝑎, where 𝑎=12, 𝑎=1, 𝑎=32, 𝑎=2. We will first test whether this sequence is arithmetic by performing the following calculations: 𝑎𝑎=112=12,𝑎𝑎=321=12,𝑎𝑎=232=12. In this case, we can see that there is a common difference of 𝑑=12, given that 𝑎𝑎=12 for all 𝑛. Hence, this sequence is arithmetic.
  2. This sequence will be labeled as 𝑏, where 𝑏=12, 𝑏=13, 𝑏=14, 𝑏=15. We can first test whether this sequence is arithmetic by examining the differences between the first few times. In this case, we find that 𝑏𝑏=1312=16,𝑏𝑏=1413=112,𝑏𝑏=1514=120. Even from the first two calculations, it is clear that this sequence is not arithmetic because it is not the case that 𝑏𝑏=𝑑 for all 𝑛. We will now investigate whether the sequence is geometric by completing the calculations 𝑏𝑏==23,𝑏𝑏==34,𝑏𝑏==45. This sequence is not geometric because it is not the case that 𝑏𝑏=𝑟 for all 𝑛. Therefore, this sequence is neither arithmetic nor geometric.
  3. We denote this sequence by 𝑐, with the values 𝑐=12, 𝑐=14, 𝑐=18, 𝑐=116. Since the denominators of these terms relate to the powers of two, we suspect that this is a geometric sequence. To test this hypothesis, we compute the following: 𝑐𝑐==12,𝑐𝑐==12,𝑐𝑐==12. In this case, we see that these terms do form a geometric sequence with a common ratio 𝑟=12.
  4. This sequence will be labeled as 𝑑, and the first four terms are 𝑑=13, 𝑑=13, 𝑑=1, 𝑑=53. It appears as though the sequence may be arithmetic, so we will complete the following calculations: 𝑑𝑑=1313=23,𝑑𝑑=113=23,𝑑𝑑=53(1)=23. This sequence is arithmetic, given that 𝑑𝑑=23 for all 𝑛.
  5. It may appear that this sequence is arithmetic upon inspection of the first three terms. Denoting the sequence by 𝑒, we find that 𝑒=1, 𝑒=12, 𝑒=0, 𝑒=12. Assuming that the sequence is arithmetic, we calculate the following: 𝑒𝑒=121=12,𝑒𝑒=012=12,𝑒𝑒=120=12. This sequence is also arithmetic, given that 𝑒𝑒=12 for all 𝑛.

This means that the correct option is (B), since this is the only sequence that is neither arithmetic nor geometric.

The range of sequences in the example above should give a fair indication as to how these types of problems can be approached. Generally speaking, we can usually make a reasonable assumption as to the underlying nature of a sequence by looking at the first few terms and then making a guess as to whether it is arithmetic, geometric, or neither. Once we have made our guess, we can inspect terms to see whether our prediction is correct. We will give one more example of this in the following question.

Example 2: Identifying the Type of Sequence When given the Terms

What kind of sequence is the following: 1,110,1100,11,000,?

  1. Neither geometric nor arithmetic
  2. Geometric and arithmetic
  3. Geometric only
  4. Arithmetic only

Answer

The denominators in this sequence are clearly related to powers of 10 and so we assume with a reasonable certainty that this sequence is geometric. We denote the sequence by 𝑇 and hence label the first four terms as 𝑇=1, 𝑇=110, 𝑇=1100, 𝑇=11000. We can verify that this sequence is geometric by calculating 𝑇𝑇 for all 𝑛 and seeing whether this results in a common ratio. The calculations are 𝑇𝑇=1=110,𝑇𝑇==110,𝑇𝑇==110.

As we expected, this sequence is geometric because 𝑇𝑇=𝑟 for all 𝑛, where the common ratio is 𝑟=110

The question above referred to a sequence where there was power-like behavior in the denominators of the terms. Generally speaking, if power-like behavior is occurring in a sequence, then it is more likely to be related to a geometric sequence rather than an arithmetic one. This is not a foolproof guideline for spotting geometric sequences, as often the precise behaviors of these sequences can be hidden if the numerator or denominator terms are particularly intricate.

So far in this explainer, we have given little thought as to the domain and range of a sequence. We should be familiar with the domain and range of a function, where the domain refers to the set of “input” values and the range refers to the set of “output” values. When dealing with continuous functions, we are often interested in what the range of values may be given a certain domain. Typically, with functions, we are interested in using a domain that has infinitely many, continuous values taken over some interval of the real numbers (which can of course include all of the real numbers). When we are working with sequences instead of functions, the domain and range must instead be sets of discrete values. Often, it can be helpful to approach the problem diagrammatically, and doing this allows us to use our understanding of functions in terms of their domain, range, and graphs. We will demonstrate this in the following example.

Example 3: Understanding the Range of an Infinite Sequence Represented on a Graph

Find the range of the infinite arithmetic sequence represented in the figure below.

  1. {1,2,3,4,}
  2. [8,4]
  3. {4,0,4,8}
  4. {4,0,4,8,}

Answer

The question states that the given sequence is an infinite arithmetic sequence, which means that the range (that is, the set of all possible outputs) must also be infinite. The points on the graph are given as the coordinates (1,4),(2,0),(3,4),(4,8),, and the sequence is generated by the values of the 𝑦-coordinate in each of these. If we label the general term as 𝑇, then we would have

𝑇=4,𝑇=0,𝑇=4,𝑇=8,.(1)

At this point, it is useful to mention that arithmetic sequences are sometimes called “linear sequences,” given that their points will trace out a straight line when plotted in the Cartesian plane. In this case, we can see that the coordinates given above can be generated by the straight line equation

𝑦=84𝑥,(2)

where 𝑥 takes the values 1,2,3, and 4. This straight line can then be used to extrapolate further points within the sequence when 𝑥 is a natural number.

The range of the sequence is all possible values, and we have already been told that this sequence is infinite. To begin to answer the question, we can therefore remove all options that are finite, meaning that we instantly remove both options (C) and (D), given that these only have 2 terms and 4 terms respectively. Remaining are the options (A), (B), and (E), which we will now consider in turn.

Option (A) is referring to the entire set of real numbers, which is a continuous set of values as opposed to a discrete list. For this reason alone, clearly (A) cannot represent the range of the given sequence. We can rigorously prove this with ease by taking any real number that does not appear as part of the sequence. For example, suppose we chose the real number 3. Given the equation of the straight line in (2), there is no natural number 𝑥 that can return the output value of 3. Therefore, this number is not an element of the range of the sequence.

Even though option (B) refers to a discrete set of numbers, rather than a continuum, we can discard it using the same logic as we did for option (A) using the real number 3, or any positive integer that does not appear in the sequence. Another way of realizing this is to note that option (B) seems to represent the domain of the sequence, rather than the range.

With all other options eliminated, it must be the case that option (E) is the correct choice. This matches with the terms of the sequence given in equation (1), and the ellipsis indicates that the sequence continues infinitely.

The domain of a sequence are the values that are used to generate the range. These values must be a discrete list of integers, with the first term for a sequence 𝑇 being conventionally labeled as 𝑇. This does not always have to be the case, and the terms of a sequence can be generated using any starting and finishing values that are suitable. However, all elements of the domain must be integers and separated by the same increment of 1, meaning that the positive integers are the natural choice of domain in most circumstances.

Example 4: Understanding the Domain and Range of an Infinite Sequence Represented on a Graph

Consider the finite sequence {20,14,8,2,4,,,22}, where two values are missing. We can think of this sequence as the function whose graph is sketched.

  1. What is the domain of the function?
  2. What is the range of the function?

Answer

Part 1

The domain is the set of all possible inputs. We are clearly working with a linear sequence, meaning that the output values can be represented by the straight line equation 𝑦=𝑚𝑥+𝑐. From the plot of the graph, we can deduce that every point has a positive 𝑥-coordinate. Furthermore, the spacing between the 𝑦-axis and the leftmost coordinate appears to be the same as between the plotted points. This suggests that the 𝑥-values of the coordinates take the form 𝑐,2𝑐,3𝑐,4𝑐,5𝑐,6𝑐,7𝑐,8𝑐, where 𝑐 is some positive number. Slightly separately, we know that sequences are typically specified by the first element, followed by their second element, followed by their third element, and so on. In other words, for a sequence 𝑇, we would denote the elements by 𝑇,𝑇,𝑇, and so on. Taken together, this suggests that the domain of the sequence is {1,2,3,4,5,6,7,8}.

Part 2

We can use the (now) known coordinates to find the equation of the straight line. Given that the domain is now known, we can write the coordinates out as follows: (1,20),(2,14),(3,8),(4,2),(5,4),(6,),(7,),(8,22).

We can use any two of these coordinates to derive the equation for the straight line. For example, suppose that we choose the two coordinates (1,20) and (2,14) that we label, respectively, as (𝑥,𝑦) and (𝑥,𝑦). There are multiple ways of calculating the equation of this straight line, but one way is by solving 𝑦𝑦𝑦𝑦=𝑥𝑥𝑥𝑥 for 𝑦. Substituting in the given values returns 𝑦201420=𝑥121 and solving for 𝑦 gives

𝑦=266𝑥.(3)

We can see that this straight line, when 𝑥 is a natural number, will return the coordinates of the elements of the sequence that we have been given in the statement of the question. We have already established that the domain of the function is {1,2,3,4,5,6,7,8} and we can now choose to substitute elements of this set into the straight line specified in equation (3). For example, substituting 𝑥=1 into this equation gives 𝑦=20, hence the coordinate (1,20). Similarly, substituting 𝑥=2 gives 𝑦=14, hence the coordinate (2,14). We can continue this process to replicate all elements of the sequence that were specified in the question.

The remaining unknown elements of the sequence are the sixth and seventh entries, which can now be generated using equation (3). First we substitute 𝑥=6 to give 𝑦=10 and then we substitute 𝑥=7 to give 𝑦=16. This gives the two coordinates (6,10) and (7,16) respectively. With these elements now calculated, the range of the sequence is clearly {20,14,8,2,4,10,16,22}.

As we saw in the first of the two previous examples, a useful step with such questions is often to transpose the given set of points into their exact coordinates. In the case of an arithmetic sequence, we would expect the graph of the coordinates to mark points of a straight line, which must be the case since every successive term of an arithmetic sequence differs from the previous term by the addition of a fixed common difference. This means that the “slope” of the function is constant from point to point, which implies that the points lay on a straight line. This is different from a geometric sequence, where each term differs from the previous term after multiplication of a common ratio. If the terms of such a sequence were plotted as coordinates, then the shape would be that of an exponential graph (providing that the ratio term is positive and not equal to 1).

Example 5: Understanding Sequences That Are Represented as Patterns

Consider the following pattern.

  1. Which of the following sequences represents the number of solid blue triangles in each successive term of the pattern?
    1. 2,8,26,80,
    2. 1,3,9,27,
    3. 2,6,18,54,
    4. 2,4,12,36,
    5. 2,4,8,16,
  2. Which type of sequence is found when counting the number of solid blue triangles in the above pattern?

Answer

Part 1

By inspection, we can see that the first term of the sequence has 2 triangles, the second term has 6 triangles, the third term has 18 triangles, and the fourth term has 54 triangles. Therefore, the correct option for the first part of the sequence is option (C), which is the sequence 2,6,18,54,. We will choose to denote this sequence by 𝑇, with the first terms 𝑇=2, 𝑇=6, 𝑇=18, and 𝑇=54.

Part 2

Clearly this sequence is not arithmetic, as the difference between each term increases reading from left to right. This behavior implies some form of exponential growth, which would indicate that we are working with a geometric sequence. To probe this suggestion, we will perform the following calculations: 𝑇𝑇=62=3,𝑇𝑇=186=3,𝑇𝑇=5418=3.

These calculations have shown that 𝑇𝑇=𝑟 for all possible values of 𝑛, where 𝑟=3 is the common ratio. This confirms that the sequence is geometric.

In this explainer, we briefly discussed the properties of arithmetic and geometric sequences; these concepts will appear frequently when working with any type of sequence. Sometimes, we are interested in adding together all individual terms of such sequences, which are referred to then as arithmetic and geometric “series.” Both of these concepts appear seemingly ubiquitously across many areas of math and science. Being able to recognize and classify both types of sequences is therefore a critical skill in the toolkit of any mathematician, the wide-ranging applicability of which should never be understated!

Key Points

  • A sequence 𝑇 is arithmetic if 𝑇𝑇=𝑑 for all 𝑛. The value of 𝑑 is called the common difference.
  • A sequence 𝑇 is geometric if 𝑇𝑇=𝑟 for all 𝑛. The value of 𝑟 is called the common ratio, or simply the ratio term.
  • If the successive terms of a sequence are such that they alternate in their sign, then this sequence cannot be arithmetic. The sequence is more likely to be geometric, although this is not guaranteed and further calculations will probably still be necessary.
  • If the numerator or denominator terms of the sequence appear to exhibit a power-like behavior, then this is unlikely to be an arithmetic sequence and is more likely to be a geometric one.
  • The domain of a sequence is normally the set of positive integers.
  • When plotted against the domain values, the shape of the coordinates will indicate whether the sequence is arithmetic or geometric. If the shape of the coordinates is a straight line, then the sequence is likely to be arithmetic, and if the shape is exponential, then the sequence is likely to be geometric (with a positive ratio term that is not equal to 1).

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