# Lesson Video: Introduction to Sequences Mathematics

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

18:00

### Video Transcript

In this video, we will learn how to identify a sequence and some of the common properties of sequences. We recall that a sequence in mathematics is an ordered list of terms. In this video, we will focus on numeric sequences. Common sequences include the integers one, two, three, four, and so on, the square numbers one, four, nine, 16, and so on, together with many other recognizable ones. The order of a sequence is important. For example, one, four, nine, 16 is not the same as one, nine, four, 16. This is a key distinction between a sequence of numbers and the set of numbers.

In this video, we will focus on two types of sequences, arithmetic and geometric. An arithmetic sequence is one in which each term can be obtained from the previous one by adding a common difference 𝑑. In other words, a sequence is arithmetic if 𝑎 sub 𝑛 plus one is equal to 𝑎 sub 𝑛 plus 𝑑 where 𝑎 sub 𝑛 is the general term and 𝑛 is any natural number. 𝑎 sub 𝑛 plus one will be the term that follows this. By subtracting 𝑎 sub 𝑛 from both sides of this equation, this can be rewritten as shown.

Let’s consider the sequence three, nine, 15, 21, and so on. The first term in this sequence denoted 𝑎 sub one is equal to three, 𝑎 sub two is equal to nine, 𝑎 sub three 15, and 𝑎 sub four is 21. We can prove that this sequence is arithmetic by considering the difference between pairs of consecutive terms. Subtracting the first term from the second term, we have nine minus three, which is equal to six. 𝑎 sub three minus 𝑎 sub two is also equal to six. When we subtract the third term, 𝑎 sub three, from the fourth term, 𝑎 sub four, we also get an answer of six. The sequence three, nine, 15, 21 has a common difference of six and is therefore arithmetic. It is important to note that the common difference could be negative, in which case each term of the sequence would be less than the previous one.

We will now look at the definition of a geometric sequence. A geometric sequence is one in which each term can be obtained from the previous term by multiplying by a common ratio 𝑟. In other words, a sequence is geometric if 𝑎 sub 𝑛 plus one is equal to 𝑎 sub 𝑛 multiplied by 𝑟. Dividing through by 𝑎 sub 𝑛, this can also be written as 𝑎 sub 𝑛 plus one divided by 𝑎 sub 𝑛 is equal to the common ratio 𝑟. Let’s consider the sequence three, six, 12, 24, and so on. Once again, we will let each of these terms be 𝑎 sub one, 𝑎 sub two, 𝑎 sub three, and 𝑎 sub four, respectively. Dividing 𝑎 sub two by 𝑎 sub one, we have six divided by three, which is equal to two. When we divide the third term by the second term, we also get an answer of two.

Finally, we also get an answer of two when dividing 𝑎 sub four by 𝑎 sub three. This means that we have a common ratio equal to two and the sequence three, six, 12, 24, and so on is geometric. When dealing with arithmetic sequences, we saw that each successive term is either larger or smaller than the previous term. This will depend on whether the common difference is positive or negative. A geometric sequence, on the other hand, can alternate between positive and negative values. This occurs when the common ratio 𝑟 is negative. For example, the sequence three, negative six, 12, negative 24, and so on has a common ratio equal to negative two. In our first example, we will consider whether a list of sequences are arithmetic, geometric, or neither.

Which of the following sequences cannot be classified as arithmetic or geometric? Is it (A) one-half, one, three over two, two, and so on? (B) One-half, one-third, one-quarter, one-fifth, and so on. (C) One-half, one-quarter, one-eighth, one sixteenth, and so on. (D) One-ninth, negative one-third, one, negative three, and so on. (E) One, one-third, negative one-third, negative four-thirds, and so on.

We will begin by recalling what we mean by an arithmetic and geometric sequence. However, it is important to note that we are looking for the sequence or sequences that are not either of these. A sequence is arithmetic if 𝑎 sub 𝑛 plus one minus 𝑎 sub 𝑛 is equal to 𝑑. In other words, the difference between consecutive terms is equal to a common difference. Let’s now consider whether any of the options listed satisfy this property. Subtracting the first term from the second term in option (A) gives us one-half. This is also true when we subtract the second term from the third term and when we subtract the third term from the fourth term. We can therefore conclude that the sequence one-half, one, three over two, two has a common difference of one-half and is therefore an arithmetic sequence.

Substituting in the consecutive terms of option (B), we see that there is no common difference. This means this is not an arithmetic sequence. The same is true for options (C), (D), and (E). However, it is worth looking at option (E) more closely. Subtracting the first term from the second term here gives us negative two-thirds, and this is also true when we subtract the second term from the third term. However, subtracting the third term from the fourth term does not give us negative two-thirds. This highlights that it is important to check all the pairs of successive terms.

We will now recall our definition of a geometric sequence. Any sequence is said to be geometric if 𝑎 sub 𝑛 plus one divided by 𝑎 sub 𝑛 is equal to a common ratio 𝑟. The quotient of successive terms must be equal. Dividing the consecutive terms of option (B), we have one-third divided by one-half, one-quarter divided by one-third, and one-fifth divided by one-quarter. Recalling that dividing by a fraction is the same as multiplying by its reciprocal, we have answers of two-thirds, three-quarters, and four-fifths. This means that option (B) does not have a common ratio and is therefore not geometric. We can therefore conclude that option (B) is a correct answer. It cannot be classified as arithmetic or geometric.

Before finishing this question, it is important we check whether options (C), (D), and (E) represent geometric sequences. Dividing the consecutive terms of option (C) gives us a common ratio equal to one-half. This means that this sequence is geometric and is therefore not a correct answer. Dividing the consecutive terms of option (D) also gives us a common ratio, this time equal to negative three. The sequence one-ninth, negative one-third, one, negative three is geometric. Finally, dividing consecutive terms of option (E), we see there is no common ratio. This means that this sequence is not geometric, and we have already established it is not arithmetic. The two sequences one-half, one-third, one-quarter, one-fifth, and so on and one, one-third, negative one-third, negative four-thirds, and so on cannot be classified as arithmetic or geometric.

In our next example, we will consider the domain and range of a sequence. Let’s firstly recall what we mean by these terms. The domain of a function refers to the set of input values whereas the range refers to the set of output values. Unlike functions, when working with sequences, the domain and range must be sets of discrete values. In the example that follows, this will be demonstrated on a graph.

Find the range of the infinite arithmetic sequence represented in the figure. Is it (A) the set of all real values? (B) The set of numbers one, two, three, four, and so on. (C) The values on the closed interval negative eight to four. (D) The set of numbers four, zero, negative four, negative eight. Or (E) the set of numbers four, zero, negative four, negative eight, and so on.

We are told in the question that the given sequence is arithmetic. We are also told that it is infinite, which means that the range must also be infinite. We can therefore rule out option (C) and (D) as these contain a finite set of values. The four points shown in the figure have coordinates one, four, two, zero, three, negative four, and four, negative eight. We know that the range of a function is the set of outputs or 𝑦-values. In this case, they’re equal to four, zero, negative four, and negative eight, the values of 𝑇 sub 𝑛. The range of the infinite arithmetic sequence represented in the figure is four, zero, negative four, negative eight, and so on. This means that the correct answer is option (E).

Option (B) the set of values one, two, three, four, and so on refers to the domain as this is the set of inputs or 𝑥-values. When dealing with a sequence, we know that the range must be a discrete set of values. As option (A), the set of real numbers, is continuous, we can rule out this option. This confirms that option (E) is the correct choice.

In our final example, we will look at a sequence that is represented by a pattern.

Consider the following pattern. Which of the following sequences represents the number of solid blue triangles in each successive term of the pattern? Is it (A) two, eight, 26, 80, and so on? (B) One, three, nine, 27, and so on. (C) Two, six, 18, 54, and so on. (D) Two, four, 12, 36, and so on. Or (E) two, four, eight, 16, and so on. Which type of sequence is found when counting the number of solid blue triangles in the above pattern?

In this question, we’re interested in the number of solid blue triangles in each term. We are given five possible sequences that represents this. In term 1, it is clear that there are two blue triangles. This immediately rules out option (B) as the first term in this sequence is one. In the second term, there are six blue triangles. This rules out option (A), option (D), and option (E) as these have a second term of eight, four, and four, respectively. So far, the first two terms match those in option (C). In pattern 3, each of the sections circled has three blue triangles. As there are six of these, this gives a total of 18 blue triangles. This once again corresponds to the third term in option (C).

In term 4, each of the sections circled has nine blue triangles, giving a total of 54. The sequence that represents the number of solid blue triangles is two, six, 18, 54, and so on. We can therefore conclude that the correct answer is option (C).

In the second part of this question, we are asked to work out which type of sequence is shown. This could potentially be an arithmetic sequence, a geometric sequence, or neither. We recall that an arithmetic sequence has a common difference between consecutive terms. This is clearly not the case for this sequence. A geometric sequence has a common ratio between successive terms. As two multiplied by three is six, six multiplied by three is 18, and 18 multiplied by three is 54, the sequence two, six, 18, 54 has a common ratio equal to three. We can therefore conclude that the type of sequence that is found when counting the number of solid blue triangles in the pattern is a geometric sequence.

We will now summarize the key points from this video. A sequence 𝑎 sub 𝑘 is arithmetic if 𝑎 sub 𝑛 plus one minus 𝑎 sub 𝑛 is equal to 𝑑 for all natural numbers 𝑛. The value 𝑑 is known as the common difference. A sequence 𝑎 sub 𝑘 is geometric if 𝑎 sub 𝑛 plus one divided by 𝑎 sub 𝑛 is equal to 𝑟 for all natural numbers 𝑛 where 𝑟 is the common ratio of the sequence. Sequences can be arithmetic, geometric, or neither. The domain of a sequence is the discrete set of input values, normally, the set of positive integers, whereas the range of a sequence is the discrete set of output values. When these coordinates are plotted on a graph, its shape can help identify whether the sequence is arithmetic or geometric.