Lesson Video: Geometric Mean Mathematics • Second Year of Secondary School

Video Transcript

In this video, we’ll 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 𝑎 and 𝑏, 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 𝑎 and 𝑏 with the same sign, their geometric mean is the square root of 𝑎 times 𝑏. Notice that if the signs of 𝑎 and 𝑏 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, let’s 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, it’s six divided by three or 12 divided by six and so on. But either way, we find the common ratio is two. Now, let’s demonstrate what happens by taking the geometric mean of the first and the third term of the sequence.

Remember, if 𝑎 and 𝑏 have the same sign, their geometric mean is the square root of their product. So, in this case, that’s 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. Let’s try this again. Let’s find the geometric mean of the third and fifth terms. It’s the square root of 12 times 48, and that’s equal to 24. And once again, that’s 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, let’s just begin by finding the geometric mean of a pair of numbers.

Find the geometric mean of 16 and four.

Remember, if two numbers 𝑎 and 𝑏 have the same sign, then their geometric mean is the square root of 𝑎 times 𝑏. Our two numbers are 16 and four, and they’re both positive. So, let’s let 𝑎 be equal to 16 and 𝑏 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.

We’ve demonstrated how to find the geometric mean of two numbers. So, let’s now see how to extend that to find the geometric mean for two algebraic expressions.

Find the geometric mean of nine 𝑥 to the 36th power and 36𝑦 to the 40th power.

Remember, if 𝑎 and 𝑏 are two numbers which have the same sign, then their geometric mean is the square root of 𝑎 times 𝑏. Now, if the numbers have different signs, then the product of 𝑎 and 𝑏 is negative, and so the geometric mean is undefined. So, let’s take a closer look at the two algebraic expressions we have. The first is the product of two positive numbers. We know this because 𝑥 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. 𝑦 to the 40th power has an even power, and so it’s going to be nonnegative.

So, we can simply substitute the expressions nine 𝑥 to the 36th power and 36 𝑦 to the 40th power in for 𝑎 and 𝑏, respectively. And so, the geometric mean is the square root of nine 𝑥 to the 36th power times 36𝑦 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 it’s equal to the square root of nine times the square root of 𝑥 to the 36th power times the square root of 36 times the square root of 𝑦 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 𝑥 to the 36th power? Well, of course, the square root of some real number — let’s call that 𝑐 — can be written as 𝑐 to the power of one-half. Then, we’re finding 𝑥 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 𝑥 to the 36th power is 𝑥 to the 18th power. We repeat this with the square root of 𝑦 to the 40th power. It’s 𝑦 to the 40th power to the power of a half. And then, 40 times one-half is 20. So, the square root of 𝑦 to the 40th power is 𝑦 to the 20th power. And so, we have our geometric mean; it’s 18𝑥 to the 18th power times 𝑦 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 𝑎, 𝑏, and 𝑐. 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 that’s 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. Let’s extend this to 𝑛 numbers.

Given 𝑛 numbers 𝑎 sub one, 𝑎 sub two all the way through to 𝑎 sub 𝑛, the geometric mean is given by the 𝑛th root of the product of these 𝑛 numbers, where the product must be positive if 𝑛 is an even integer. And with this in mind, let’s 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 𝑎, 𝑏, and 𝑐, their geometric mean is the cubed root of 𝑎𝑏𝑐, 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, we’re 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 we’re 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 𝑥 is the same as 𝑥 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, we’ve considered how to find the geometric mean of two or more numbers. We’re now going to introduce a related concept known as 𝑛 geometric mean between any two numbers. Given a pair of numbers, 𝑎 and 𝑏, 𝑛 geometric means between 𝑎 and 𝑏 are the values in a geometric sequence from 𝑎 to 𝑏 with exactly 𝑛 terms in between. So, we see this is very different to finding the geometric mean of two numbers. That’s just one single number, whereas this is a sequence of 𝑛. In fact, if we consider a geometric sequence with 𝑛 terms, the number of geometric means between the first and final term is 𝑛 minus two. So, let’s 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, 𝑛 geometric means between them are the 𝑛 terms of a geometric sequence between the two given numbers. We’re looking to find five means between 21 over 38 and 672 over 19. So, we want to find a geometric sequence, and it’s 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, we’re going to use the formula that helps us find any term of a geometric sequence. It’s 𝑎 sub 𝑛 is equal to 𝑎 sub one times 𝑟 to the power of 𝑛 minus one, where 𝑟 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, 𝑎 sub one, is 21 over 38, and the seventh term, 𝑎 sub seven, is 672 over 19. This means we can generate an expression in terms of 𝑟 for the seventh term using the first. It’s 𝑎 sub seven is 21 over 38 times 𝑟 to the power of seven minus one, which in turn can be written as 21 over 38 times 𝑟 to the sixth power.

Then, we know that 𝑎 sub seven is 672 over 19. So, we can solve this equation for 𝑟 by dividing both sides by 21 over 38. That gives us 𝑟 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 𝑟 is equal to positive or negative two. But remember, we’re trying to find positive geometric means. This means our sequence itself needs to contain only positive terms. And so, we choose 𝑟 is equal to positive two.

Now, we could either substitute 𝑟 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, we’d 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 𝑟 equals two, but the other had a negative ratio 𝑟 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 𝑛 geometric means between two numbers when the number of geometric means is odd. Let’s look at one final example.

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

Remember, 𝑛 geometric means between two numbers are the 𝑛 terms of a geometric sequence between the two given numbers. So, we’re 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 𝑛th term of a geometric sequence with first term 𝑎 sub one and common ratio 𝑟. It’s 𝑎 sub 𝑛 equals 𝑎 sub one times 𝑟 to the power of 𝑛 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 𝑟; it’s two times 𝑟 to the power of five minus one or two times 𝑟 to the fourth power. So, we now have an equation which we can solve to find the value of 𝑟.

We begin by dividing both sides of this equation by two. So, 𝑟 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 𝑟 as positive or negative seven. And so, in fact, there are two possible sequences we’re interested in. The first is when 𝑟 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 𝑟 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.

Let’s 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 𝑛 numbers, 𝑎 sub one through to 𝑎 sub 𝑛, the geometric mean is the 𝑛th root of the product of these. But if 𝑛 is an even integer, then their product must be positive. We saw that 𝑛 geometric means between a pair of numbers are the values in a geometric sequence between those two numbers where there are exactly 𝑛 terms between them. And then it also makes sense that if we’re given a geometric sequence with 𝑛 terms, there must be 𝑛 minus two terms between the first and the last of these.