### Video Transcript

In this video, weβll learn how to
find geometric means between two nonconsecutive terms of a geometric sequence.

When we think about the mean of two
numbers, we think about the arithmetic mean. So, if we take two numbers π and
π, we add those numbers together and divide by two. But this, in fact, is not the only
notion of the mean. For instance, the geometric mean of
two numbers with the same sign is defined as the square root of the product of the
two numbers. Formally, we say, given a pair of
numbers π and π with the same sign, their geometric mean is the square root of π
times π. Notice that if the signs of π and
π are different, their product will be negative, and so the geometric mean is
undefined.

Now, we also think about the
geometric mean of two numbers in relation to geometric sequences. Remember, a geometric sequence is a
sequence of numbers that has a common ratio between successive terms. For instance, letβs take the
sequence with terms three, six, 12, 24, and 48. The ratio is found by dividing any
term by the term that precedes it. So, itβs six divided by three or 12
divided by six and so on. But either way, we find the common
ratio is two. Now, letβs demonstrate what happens
by taking the geometric mean of the first and the third term of the sequence.

Remember, if π and π have the
same sign, their geometric mean is the square root of their product. So, in this case, thatβs the square
root of three times 12 or the square root of 36, which is, of course, equal to
six. And then, we notice that this is
equal to the second term in our sequence. So, the geometric mean of the first
and third term is the value of the second. Letβs try this again. Letβs find the geometric mean of
the third and fifth terms. Itβs the square root of 12 times
48, and thatβs equal to 24. And once again, thatβs the term
that lies between our two terms. So, the geometric mean of the third
and fifth terms is the fourth term.

We can generalize in the following
way. Given a geometric sequence with a
positive ratio, any intermediate term of the sequence is the geometric average or
mean of the two neighboring terms. So, with all of these definitions
in mind, letβs just begin by finding the geometric mean of a pair of numbers.

Find the geometric mean of 16 and
four.

Remember, if two numbers π and π
have the same sign, then their geometric mean is the square root of π times π. Our two numbers are 16 and four,
and theyβre both positive. So, letβs let π be equal to 16 and
π be equal to four. Their geometric mean then is the
square root of 16 times four. And whilst we can apply the laws of
radicals to simplify this, in fact 16 times four is 64, which is itself a square
number. So, since the square root of 64 is
equal to eight, the geometric mean of 16 and four is eight.

Weβve demonstrated how to find the
geometric mean of two numbers. So, letβs now see how to extend
that to find the geometric mean for two algebraic expressions.

Find the geometric mean of nine π₯
to the 36th power and 36π¦ to the 40th power.

Remember, if π and π are two
numbers which have the same sign, then their geometric mean is the square root of π
times π. Now, if the numbers have different
signs, then the product of π and π is negative, and so the geometric mean is
undefined. So, letβs take a closer look at the
two algebraic expressions we have. The first is the product of two
positive numbers. We know this because π₯ to the 36th
power has an even power, so substituting any real number into this expression will
give a positive output. And our next is also the product of
two positive numbers. π¦ to the 40th power has an even
power, and so itβs going to be nonnegative.

So, we can simply substitute the
expressions nine π₯ to the 36th power and 36 π¦ to the 40th power in for π and π,
respectively. And so, the geometric mean is the
square root of nine π₯ to the 36th power times 36π¦ to the 40th power. And we could at this stage multiply
nine and 36 and then the algebraic expression. But the product of nine and 36 is
quite a big number. So, instead, we can use the laws of
radicals to separate each expression. And when we do, we see that itβs
equal to the square root of nine times the square root of π₯ to the 36th power times
the square root of 36 times the square root of π¦ to the 40th power. Then we know that the square root
of nine is three, and the square root of 36 is six. Three times six is 18, so the
coefficient of our final expression is going to be 18.

But how do we evaluate the square
root of π₯ to the 36th power? Well, of course, the square root of
some real number β letβs call that π β can be written as π to the power of
one-half. Then, weβre finding π₯ to the 36th
power to the power of a half. And to simplify this, we multiply
the exponents; 36 times one-half is 18. So, the square root of π₯ to the
36th power is π₯ to the 18th power. We repeat this with the square root
of π¦ to the 40th power. Itβs π¦ to the 40th power to the
power of a half. And then, 40 times one-half is
20. So, the square root of π¦ to the
40th power is π¦ to the 20th power. And so, we have our geometric mean;
itβs 18π₯ to the 18th power times π¦ to the 20th power.

In our previous two examples, we
calculated the geometric mean of a pair of numbers or expressions. In fact, this idea can be extended
to find the geometric mean of any amount of numbers. Take, for instance, three numbers
π, π, and π. Their geometric mean is the cubed
root of the product of these three numbers. And notice that unlike the
geometric mean for two numbers, the geometric mean for three is well defined even if
the signs of the numbers are different. And thatβs because the cubed root
of a negative number is defined. However, in practice, we generally
only work with positive values when calculating a geometric mean. Letβs extend this to π
numbers.

Given π numbers π sub one, π sub
two all the way through to π sub π, the geometric mean is given by the πth root
of the product of these π numbers, where the product must be positive if π is an
even integer. And with this in mind, letβs
consider an example where we calculate the geometric mean of three numbers using
this formula.

Find the geometric mean of the
numbers six, 72, and 108.

Remember, given three numbers π,
π, and π, their geometric mean is the cubed root of πππ, the product of those
three numbers. Now, our numbers are six, 72, and
108. So, we find the geometric mean by
substituting these three numbers into this expression. And so, the geometric mean is the
cubed root of six times 72 times 108.

Now, whilst we could multiply these
and then evaluate the cubed root using our calculator, weβre going to perform some
tricks to do this in our head. We begin by writing each number as
a product of its prime factors. So, six is two times three, and we
can use a factor tree if necessary. 72 is two cubed times three
squared, and 108 is two squared times three cubed. And so, we replaced the expression
inside the cubed root with two times three times two cubed times three squared times
two squared times three cubed.

Then, we notice that we can add the
exponents for the numbers whose base is the same since weβre multiplying. Since two and three are
individually raised to the power of one, we get the cubed root of two to the sixth
power times three to the sixth power. But then, we note that the cubed
root of some number π₯ is the same as π₯ to the power of one-third. And so, two to the sixth power to
the power of one-third is equivalent to two squared. Similarly, three to the sixth power
to the power of one-third is equal to three squared. This, in turn, is equal to four
times nine, which is equal to 36. So, the geometric mean of the three
numbers six, 72, and 108 is 36.

Up to this stage, weβve considered
how to find the geometric mean of two or more numbers. Weβre now going to introduce a
related concept known as π geometric mean between any two numbers. Given a pair of numbers, π and π,
π geometric means between π and π are the values in a geometric sequence from π
to π with exactly π terms in between. So, we see this is very different
to finding the geometric mean of two numbers. Thatβs just one single number,
whereas this is a sequence of π. In fact, if we consider a geometric
sequence with π terms, the number of geometric means between the first and final
term is π minus two. So, letβs demonstrate this in the
next example.

Insert five positive geometric
means between 21 over 38 and 672 over 19.

Remember that if we have a pair of
numbers, π geometric means between them are the π terms of a geometric sequence
between the two given numbers. Weβre looking to find five means
between 21 over 38 and 672 over 19. So, we want to find a geometric
sequence, and itβs going to be positive, as per the question, that begins with 21
over 38, ends with 672 over 19, and has exactly five terms in between.

So, weβre going to use the formula
that helps us find any term of a geometric sequence. Itβs π sub π is equal to π sub
one times π to the power of π minus one, where π is the common ratio. For there to be five terms between
the first and the last, that must mean that there are seven terms altogether. So, the first term, π sub one, is
21 over 38, and the seventh term, π sub seven, is 672 over 19. This means we can generate an
expression in terms of π for the seventh term using the first. Itβs π sub seven is 21 over 38
times π to the power of seven minus one, which in turn can be written as 21 over 38
times π to the sixth power.

Then, we know that π sub seven is
672 over 19. So, we can solve this equation for
π by dividing both sides by 21 over 38. That gives us π to the sixth power
is equal to 64. Then, we can solve this equation by
taking the positive and negative sixth root of 64, giving us that π is equal to
positive or negative two. But remember, weβre trying to find
positive geometric means. This means our sequence itself
needs to contain only positive terms. And so, we choose π is equal to
positive two.

Now, we could either substitute π
is equal to two into our earlier formula, or we can use the fact that to generate
each term in a sequence, we multiply it by the common ratio. So, the second term is the first
term times two, which is 21 over 19. The third term is the second term
times two, which is 42 over 19. We keep going in this way, giving
us a fourth term of 84 over 19, a fifth term of 168 over 19, and a sixth term of 336
over 19. And in fact, if we then multiply
this value by two, weβd get 672 over 19, as we expected. So, our five positive geometric
means are 21 over 19, 42 over 19, 84 over 19, 168 over 19, and 336 over 19.

Now, in this example, we looked at
geometric means between a given pair of numbers. But remember, there were two
possible sequences we could have generated. The first, the one we looked at,
had the positive ratio π equals two, but the other had a negative ratio π equals
negative two. And so, what this tells us is that
unless the geometric means are restricted to a specific sign, we can generate two
sets of π geometric means between two numbers when the number of geometric means is
odd. Letβs look at one final
example.

Find the geometric means of the
sequence whose first term is two and whose last term is 4802.

Remember, π geometric means
between two numbers are the π terms of a geometric sequence between the two given
numbers. So, weβre going to need to identify
the sequence whose first term is two and whose last term is 4802 and where there are
exactly three terms between them. So, we use the formula for the πth
term of a geometric sequence with first term π sub one and common ratio π. Itβs π sub π equals π sub one
times π to the power of π minus one.

The first term in our sequence is
two; then the fifth term is 4802. But then using our formula, we can
express that in terms of π; itβs two times π to the power of five minus one or two
times π to the fourth power. So, we now have an equation which
we can solve to find the value of π.

We begin by dividing both sides of
this equation by two. So, π to the fourth power is
2401. Then, we can find the positive and
negative fourth root of 2401. And remember, we do this because
four is an even exponent or power, and this gives us a value for π as positive or
negative seven. And so, in fact, there are two
possible sequences weβre interested in. The first is when π is equal to
seven. Beginning with our first term two,
we multiply by seven each time. And this gives us the sequence two,
14, 98, 686, and 4802. And if π is negative seven, we
change the sign of the 14 and the 686. So, the geometric means of this
sequence are either 14, 98, 686 or negative 14, 98, and negative 686.

Letβs now recap the key points from
this lesson. In this lesson, we learned that we
can find the geometric mean of any amount of numbers. Given π numbers, π sub one
through to π sub π, the geometric mean is the πth root of the product of
these. But if π is an even integer, then
their product must be positive. We saw that π geometric means
between a pair of numbers are the values in a geometric sequence between those two
numbers where there are exactly π terms between them. And then it also makes sense that
if weβre given a geometric sequence with π terms, there must be π minus two terms
between the first and the last of these.