Question Video: Finding Geometric Mean of Two Algebraic Expressions Mathematics • Second Year of Secondary School

Video Transcript

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.