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
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.