### Video Transcript

In this video, we will learn about
geometric sequences. We will discover how to find the
general or πth term and how to find the term-to-term rule. Weβll also cover how to find the
order of a term given its value.

The first thing to note about a
geometric sequence is these are sequences where the ratio between terms is
constant. Notice that thatβs different to
arithmetic sequences, where itβs the difference between terms thatβs constant. We can describe a geometric
sequence in one of two ways, either by using a term-to-term rule or by using a
position-to-term rule. The position-to-term rule is often
called the πth term, and itβs very helpful for finding the value of a specific
term. For example, if we wanted to find
the 15th term of a sequence, we could directly evaluate it by substituting 15 into
the πth term rule rather than having to work out all the terms up to the 15th one
using the term-to-term rule.

So, letβs think about some of the
notation that we use in geometric sequences. We say that if the first term is
denoted as the letter π and the common ratio is π, then our sequence would look
like this. The first term is π. The second term would be ππ
because weβve multiplied the first term π by the common ratio π. Multiplying the second term ππ
with another π would give us ππ squared. We can signify the terms by using
subscript notation. For example, the first term would
be written as π sub one, the second term would be written as π sub two, the third
as π sub three, and so on.

So, how could we find a rule to
find the πth term which is written as π sub π? Well, we could start by noticing
that each term has an exponent value of π which is one less than the term
number. We know that the πth term would
still have an π-value, and the exponent of π would be one less than π. We can, therefore, say that the
πth term of any geometric sequence can be written as π times π to the power of π
minus one. We remember, of course, that itβs
just the π thatβs taken to the power of π minus one and it doesnβt include the π
as well. So, letβs see how we can put this
formula into practice to find the πth term of our first geometric sequence.

Find the general term of the
geometric sequence negative 76, negative 38, negative 19, negative 19 over two.

Another way of phrasing general
term is the πth term. So, weβre looking for the πth term
of this geometric sequence, which is a sequence which has a common ratio between the
terms. So, letβs have a look at the
sequence and see what we can determine. Firstly, we can see the first term
in the sequence is negative 76. When weβre working with geometric
sequences, we usually use the letter π to signify the first term of the
sequence. In order to find the πth term,
weβll also need to find π, the common ratio. When we consider a general
geometric sequence written as π then ππ then ππ squared, and so on, we can find
the common ratio π by dividing any term by the immediately preceding term.

So, here, we could take the second
term of negative 38 and divide it by negative 76. Therefore, π is equal to
one-half. Notice that even if weβd taken two
different terms, for example, if we divided the third term of negative 19 by the
second term of negative 38, we would still have found the common ratio π to be
one-half. After all, if it wasnβt the same,
then we wouldnβt have a geometric sequence.

So now that we found the values of
π and π, we recall the general formula for the nth term of a geometric
sequence. π sub π, thatβs the πth term, is
equal to π times π to the power of π minus one. All we need to do now is plug in
the values of π equals negative six and π equals a half into this formula. This gives us π sub π equals
negative 76 times a half to the power of π minus one. As we canβt simplify this any
further, then thatβs our answer for the general term or the πth term of the
geometric sequence.

Letβs have a look at another
question where weβre finding the πth term of a slightly more complex sequence.

Find, in terms of π, the general
term of the sequence one-fourth, nine over 16, 81 over 64, 729 over 256, and so
on.

In this question, weβre given the
first four terms of this sequence. There doesnβt look like thereβs a
common difference between the terms, so we could say that this is definitely not an
arithmetic sequence. We can check if itβs a geometric
sequence which would have a common ratio between the terms by seeing if we can work
out what that common ratio would be.

In order to find the ratio π
between the first two terms, we would take the second term, nine over 16, and divide
by the term before it, one-quarter. We can recall that to divide by
one-quarter, that would be equivalent to multiplying by the reciprocal, which would
be four over one. We can simplify the four on the
numerator and the 16 on the denominator by taking out a factor of four. We then multiply our numerators and
denominators. Nine times one would give us nine,
and four times one gives us four. We can check if thereβs the same
ratio between the third term and the second term. So, we calculate 81 over 64 divided
by nine over 16. We can see that the reciprocal of
nine over 16 would be 16 over nine. And we simplify the fractions
before multiplying the numerators and denominators, which gives us the same common
ratio π of nine over four. And if we check this by working out
what we need to multiply the third term 81 over 64 by in order to get the fourth
term of 729 over 256, we get the same answer of nine over four.

So, letβs think about how we would
find the general term of this sequence. We can recall that the general term
is another way of saying the πth term. We can recall that the formula to
find the πth term π sub π is that π sub π is equal to π times π to the power
of π minus one. The π-value represents the common
ratio, and the π-value represents the first term of the sequence. Weβve already established that π
is equal to nine over four, and the π-value, the first term in the sequence, would
be one-quarter. We can then take the values of π
and π and plug them into this formula. This gives us π sub π equals
one-quarter times nine-quarters to the power of π minus one.

While this is a perfectly valid
answer for the πth term of the sequence, it might be worth seeing if we can
simplify this right-hand side any further. We can start our simplification by
thinking about what happens when we have a power of a fraction. A fraction with a certain exponent
value is equivalent to the numerator with that exponent over the denominator with
that exponent. If we then consider these fractions
multiplied, multiplying the numerators one times nine to the power of π minus one
would give us the value on the numerator of nine to the power of π minus one. Multiplying the denominators would
give us four times four to the power of π minus one.

If we look at the denominator, we
can apply another exponent rule that π₯ to the power of π multiplied by π₯ to the
power of π is equal to π₯ to the power of π plus π. On the denominator, the π₯-value
here would be four. And our π- and π-values, thatβs
the exponents, would be represented by one and π minus one. Adding one and π minus one would
leave us with the value of π. So, weβve found a fully simplified
answer for the general term of this sequence in terms of π. Itβs nine to the power of π minus
one over four to the power of π.

In the two questions that weβve
just seen, weβve been finding the πth term or the position-to-term rule. In the next question, weβll have a
look at how we would find a term-to-term rule.

Find a recursive formula for the
sequence 486, 162, 54, 18, six, two, two-thirds.

So here we have a sequence which
begins with the number 486, and weβre asked to find a recursive formula. This means that instead of finding
a general term or an πth term for the sequence, weβre going to find a term-to-term
rule. It means weβll be thinking about
how we go from one term to another term. Letβs begin by seeing if we can
find a common ratio between the terms. We can signify this with the letter
π. In a geometric sequence, we say
that the first term is π, the second term would be π times the common ratio π,
the third term would be ππ squared, and the fourth term would be ππ cubed, and
so on. Therefore, to find the ratio
between terms, we take any term and divide by the term immediately before it.

So, we could, for example, take the
second term of 162 and divide by the term before it, 486. Or if itβs a geometric sequence, as
we assume, then we could take the third term and divide by the second term. We could even take the sixth term
of two and divide by the fifth term. We should hopefully see
straightaway that two-sixths can simplify to one-third. So, do the other two fractions here
also simplify to one-third? Well, if we take 54 and add 54,
thatβs 108. And adding another 54 would give us
162. So, this fraction would also
simplify to one-third. 162 over 486 would also give us
one-third.

Itβs looking like we have a common
ratio of one-third. In fact, if we take any two
consecutive terms in this sequence, we find that we need to multiply the first one
by one-third to get the second. The terms we have been given
definitely do have a common ratio of one-third. So, does this answer the question
then to find a recursive formula? Is it sufficient to say we would
multiply a term by one-third to find the next one?

Well, not quite. We can use some more formal
notation here. We say that the first term π one
will be equal to 486. To find the second term, we know
that that would be 486 times the ratio π of one-third. So, π sub two, the second term, is
equal to one-third of π sub one. In the same way, if we wanted to
find the third term and we didnβt know the value of it, we take the second term and
multiply it by the ratio π of one-third. We could continue like this to say
the fourth term would be equal to one-third times the third term.

So, if we wanted to find the πth
term of this sequence, we would calculate one-third multiplied by the term before,
written as π sub π minus one. This is where we get the helpful
formula from, that if we want to find the π plus oneth term of a sequence, then we
multiply the common ratio by the πth term. It sort of looks like the formula
for the πth term of a sequence, but this time the term is based on the term before
it rather than the starting term of a sequence. But here, we can give our answer
that any term in the sequence π sub π can be calculated by a third times the term
before it, π sub π minus one.

In the next question, weβll see how
we can find the order of a term given its value and the πth term of the
sequence.

Find the order of the term whose
value is 4374 in the geometric sequence π sub π equals two-thirds times three to
the power of π.

Letβs start by looking at the
information that weβre given. This value of π sub π represents
the πth term of this sequence. Weβre asked to find the order of
the term whose value is 4374. That means that weβve got a
sequence, and somewhere in this sequence is this value of 4374. The order of this term means weβre
really asking, is it the second term, the 10th term, the 100th term? Thatβs what we need to find
out. We can do this by saying letβs make
the order of this term π, and then our πth term will be 4374. We could then fill this into the
formula and rearrange to find this value of π, which would give us the order of
this term.

We can start our rearranging by
dividing both sides of this equation by two-thirds. On the left-hand side, we can
recall that to divide by a fraction, we multiply by its reciprocal. And on the right-hand side, weβll
be left with three to the power of π. We can simplify the values on the
left-hand side. So, we work out 2187 multiplied by
three, which gives us 6561 is equal to three to the power of π.

Now, at this stage, thereβs a
branch of mathematics called logarithms, which would help us solve this problem
directly. But as most people learn this long
after they learn about geometric sequences, weβll use a bit of trial and improvement
here instead. Remember that a value like three to
the power of π equals 6561 is really equivalent to saying three to the power of
what gives us this value. You might know your first powers of
three off by heart, up to roughly three to the power of four equals 81. We could then continue with a few
more by multiplying each of the values by three as we go up. If weβre using a non-calculator
method, weβll probably need to start using some pencil and paper working out. But then, we find that three to the
power of eight is equal to 6561. This means that our π-value here
must be equal to eight. So, we can give our answer that the
order of the term whose value is 4374 is eight, as it would be the eighth term in
this sequence.

We can now summarize what weβve
learnt in this video. Firstly, we saw that a geometric
sequence is a sequence where the ratio between terms is constant. For a first term π and a common
ratio π, we have the terms of the sequence as π, ππ, ππ squared, ππ to the
third power, and so on. To find the common ratio π given a
value of terms in a sequence, we can divide any term by its preceding term. For example, π can be found by
dividing the third term by the second term. The position-to-term rule or the
πth term can be written as π sub π equals π times π to the power of π minus
one.

Finally, we saw the term-to-term
rule, or recursive formula, which can be written as π sub π plus one equals π
times π sub π, remembering that this means if we want to find the value of the
term with order π plus one, then we take the value of the term with order π and
multiply it by the common ratio π.