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.