Video Transcript
In this video on geometric
sequences, we will learn how to calculate the common ratio between terms, find the
next terms in a geometric sequence, and check if the sequence increases or
decreases.
But letβs start by thinking about
what a geometric sequence actually is. One of the geometric sequences we
might commonly see is the sequence one, two, four, eight, 16, and so on. Each term in the sequence is double
the term before it. We can say that the ratio between
any two successive terms in the sequence is two. We can see a whole range of
different geometric sequences. For example, we could have
sequences with noninteger values and sequences which have a negative ratio between
successive terms. These type of sequences are
referred to as alternating geometric sequences because the signs switch between
positive and negative.
But what defines a geometric
sequence is that the ratio is common between successive terms. We can use the terminology that the
sequence can be given as the terms π sub one, π sub two, π sub three, π sub
four, and so on. We may also see this with
alternative letters other than using π, for example, the sequence defined as π‘ sub
one, π‘ sub two, and so on. Some sequences may begin with π
sub zero, but weβre really just giving a position value to each term.
We can define a geometric sequence
then as a sequence of nonzero numbers π sub one, π sub two, π sub three, π sub
four, and so on that has a nonzero common ratio π, which is not equal to one,
between any two consecutive terms. The common ratio π is equal to π
sub π plus one over π sub π for values of π equal to one, two, three, and so
on. This part of the equation π sub π
plus one over π sub π simply indicates any term in the sequence divided by the
term immediately before it.
If we take this example sequence
below, we can say that the ratio π is equal to π sub two over π sub one. Thatβs 100 divided by negative 10,
which is negative 10. But we could have equally found the
ratio by dividing π sub four by π sub three. And that would also give us the
same ratio of negative 10. But before we look at some
questions, we can also define increasing and decreasing geometric sequences.
We say that a geometric sequence is
increasing if π sub π plus one is greater than π sub π, and it would be
decreasing if π sub π plus one is less than π sub π. So itβs increasing if each term is
greater than the term before it and decreasing if a term is less than the term
before it. In each case, this has to be true
for all index values of π.
We can now take a look at some
examples on geometric sequences, and weβll begin with one in which we need to find
the common ratio.
The table shows the number of
bacteria in a laboratory experiment across four consecutive days. The number of bacteria can be
described by a geometric sequence. Find the common ratio of this
sequence.
In the table, we can see that on
four different days, there is a number of bacteria recorded in this experiment. We are told that the number of
bacteria forms a geometric sequence and that is one which has a common ratio between
any two consecutive terms. In order to find this ratio, then,
we could take any term and divide it by the term before it. For example, we could take the
second term, which could be denoted as π sub two, and divide it by the first
term.
This would be 2572 divided by
643. Putting this into our calculators
or simplifying the fraction, we would get an answer of four.
As a check of our answer, we could
find the ratio between another pair of consecutive terms. For example, we could take the
fourth term and divide it by the third term. Simplifying 41152 over 10288 would
also give us an answer of four. Because we know that this is a
geometric sequence, we donβt need to do both sets of calculations. The second one was a good check,
however. We can give the answer, then, that
the common ratio of this sequence is four.
In the next example, weβll see how
we can find the next term of a geometric sequence.
Find the next term of the geometric
sequence negative five, negative five over four, negative five over 16, negative
five over 64, what.
Because we are given that this
sequence is geometric, that means we know that there is a common ratio between
consecutive terms. In other words, there is some ratio
π which we can multiply any term by to get the next term. And so, in order to find this value
of π, we can take any term in the sequence, which we can call π sub π plus one,
and divide it by the term before, which we can denote as π sub π.
We can take two terms in the
sequence which might be the easiest to divide. So letβs take the second term and
divide it by the first term. So we have negative five over four
over negative five. And itβs helpful to remember that
this is the same as negative five over four divided by negative five. If we take negative five as the
fraction negative five over one, then we can remember that to divide by a fraction,
we multiply by its reciprocal.
We can take out a common factor of
five and then observe that we have a negative fraction multiplied by a negative
fraction. And so weβve worked out that the
ratio π is one-quarter.
This means that to work out the
fifth term, we multiply the fourth term by the fraction of one-quarter. So we work out negative five over
64 multiplied by one-quarter. This gives us negative five over
256. We canβt simplify this fraction any
further, so we can give the answer that the next term of this geometric sequence is
negative five over 256.
In the next example, we will find
the common ratio of a geometric sequence which is defined in a recursive
formula.
Find the common ratio of the
geometric sequence which satisfies the relation π sub π equals nine-eighths π sub
π plus one, where π is greater than or equal to one.
Letβs begin by recalling that a
geometric sequence is one which has a common ratio π between any two consecutive
terms. We can find this common ratio by
dividing any term, which we denote as π sub π plus one, by the term immediately
before it, which we denote as π sub π. This is usually easy to do if weβre
actually given the terms in the sequence, but we arenβt in this question. But we are given a relationship
between π sub π and π sub π plus one.
So letβs look at this statement,
this equation that we have written regarding the common ratio. If we multiply both sides of this
equation by π sub π, we get the equation π times π sub π equals π sub π plus
one. By dividing through by π, we get
that π sub π equals π sub π plus one over π. We can then relate this to the
relation in the question, which is written in terms of π sub π. We can set the right-hand side of
both of these equations equal to each other, so we have π sub π plus one over π
equals nine-eighths π sub π plus one.
Dividing both sides of this
equation by π sub π plus one, we have one over π equals nine-eighths. And so the answer is that the
common ratio π is eight-ninths.
This question can be a little
tricky to understand, especially if weβre wondering why the ratio isnβt just nine
over eight. So letβs consider this question as
a diagram. Imagine that we have this sequence,
and we donβt know the values in the sequence. We do, however, know a relationship
between a term π sub π and π sub π plus one. But weβre almost given the
relationship in the wrong direction; weβre told how to get π sub π from π sub π
plus one. We multiply π sub π plus one by
nine over eight to get π sub π.
But when weβre thinking about
sequences, we think about how we go from one term to the term after that term. The inverse of multiplying by nine
over eight is dividing by nine over eight. But when weβre giving a ratio, it
must be in terms of a multiplier. Thatβs the reciprocal. So here we would multiply by
eight-ninths. And thatβs why the common ratio of
this sequence is eight over nine.
In the next example, weβll see how
we can generate the first terms of a sequence given its general term.
Find the first five terms of the
sequence π sub π given π sub π plus one equals one-quarter π sub π, π is
greater than or equal to one, and π sub one equals negative 27.
In this question, weβre given the
information which we can use to generate the terms of this sequence. Although this notation of π sub π
plus one and π sub π can look confusing, all that this formula is telling us is
that if we wish to generate any term in the sequence, then we take the term before
it and multiply it by one-quarter. The first five terms in the
sequence can be given as π sub one, π sub two, π sub three, π sub four, and π
sub five. We know that the sequence will
start with an index π of one because weβre told that π is greater than or equal to
one.
So letβs use this formula and say,
for example, that we wanted to work out the third term, π sub three. We can use the formula to tell us
that π sub three is one-quarter of π sub two; itβs one-quarter of the second
term. But the problem is, is that we
donβt yet know the second term of the sequence. We could work out the second term
as one-quarter of the first term, but what is the first term?
Well, weβre told that π sub one is
negative 27. In this sort of formula, which is a
recursive formula, then we need to be given at least one of the terms in order to
give us somewhere to start with the sequence. Looking at the second term, as
previously mentioned, we can find this by taking one-quarter of the first term. We can work out one-quarter
multiplied by negative 27. And negative 27 over four is the
simplest form of this fraction, and so thatβs the second term.
The third term π sub three is
one-quarter multiplied by the second term, which is one-quarter times negative 27
over four. So we have negative 27 over 16. The fourth term is one-quarter
times the third term of negative 27 over 16, which is negative 27 over 64. Finally, the fifth term is
one-quarter times negative 27 over 64, which is negative 27 over 256. We can then give the answer that
the first five terms of the given sequence are negative 27, negative 27 over four,
negative 27 over 16, negative 27 over 64, and negative 27 over 256.
In the final example, we will
consider what the graph of a geometric sequence would look like.
True or False: The terms of a
geometric sequence can be plotted as a set of collinear points.
Letβs begin this question by
recalling that a geometric sequence is a sequence which has a common ratio between
any two consecutive terms. In order to consider whether a
geometric sequence might be collinear, which means lying on a straight line, then it
might be useful to take a few examples of geometric sequences.
So letβs take the sequence one,
three, nine, 27, and so on. We can say that itβs geometric
because the common ratio is three. Any term is found by multiplying
the previous term by three. If we were to graph these, then
weβd be graphing the π or the index value alongside the term value. We could start with the coordinate
one, one because the first term has a value of one. The second coordinate would be that
of two, three. The term with index two has a value
of three.
However, a third coordinate of
three, nine might begin to reveal the pattern in this geometric sequence. We would not have a straight
line. In fact, what we would have would
be an exponential graph.
So letβs try another geometric
sequence. This time, letβs try a decreasing
geometric sequence. The sequence negative two, negative
four, negative eight, negative 16, and so on has a common ratio of two. Letβs try plotting these term
values. Once again, we can see that these
points would not lie on a straight line.
But there is another type of
geometric sequence, and thatβs an alternating sequence. The signs of the terms alternate
between positive and negative. This is because the ratio is a
negative value. When we plot this geometric
sequence, we get a graph that looks like this. So far, none of the sequences that
we have considered have created a set of collinear points. So letβs consider what type of
sequence would.
Well, if we had a sequence which
produces a straight line, that means that when the index increases, the terms
increase by a constant or a constant difference. This type of sequence is actually
an arithmetic sequence. And itβs defined by a common
difference between any two consecutive terms. Remember that a geometric sequence
has a common ratio between terms, and that common ratio cannot be equal to one. It is therefore not possible that a
geometric sequence can be plotted as a set of colinear points. That means that the statement in
the question is false.
Itβs worth noting that arithmetic
sequences are always linear but geometric sequences are never linear. In fact, they would create an
exponential function.
We can now summarize the key points
of this video. We began by considering that a
geometric sequence is a sequence of nonzero numbers with a common ratio π, which is
not equal to one, between any two consecutive terms in the sequence. We saw how we can calculate this
common ratio π by taking any term in the sequence and dividing it by the term
beforehand. This can be written as π equals π
sub π plus one over π sub π.
We also saw that a given geometric
sequence may be defined as a set of numbers π sub one, π sub two, π sub three; a
recursive formula; or an explicit formula. Finally, we saw that geometric
sequences can be increasing, decreasing, or alternating.