Question

Please help!
Find the four geometric means between 4 and 9724

Answer 1

\(4\;\;4r\;\;4r^{2}\;\;4r^{3}\;\;4r^{4}\;\;9724 = 4r^{5}\)

Answer 2

Can you explain please?

Answer 3

I just did. What's a "Geometric Mean"?

Answer 4

The geometric mean is nth root of the product of the numbers.

Answer 5

That's pretty good, but that helps you with only ONE such mean. What does it mean when there are four?

Answer 6

I don't know

Answer 7

Well, that's what I wrote. Check it out for one geometric mean between to given values.

7 _____ 23

We have established the answer as \(\sqrt{7\cdot 23} = 12.68857754\)

The expression I wrote is, for this smaller problem, equivalent to

\(7\;\;7r\;\;23 = 7r^{2}\)

The last expression leads to \(r = 1.81265393435\)

And we see:

\(7\)

\(7\cdot 1.81265393435 = 12.68857754\)

\(7\cdot (1.81265393435)^{2} = 23\)

Do you see that we managed the same result?

Multiple Geometric means are numbers in Geometric Sequence. This means they MUST differ by a common ratio. We're just finding the common ratio. Using the cute square root formula for calculating one creates the geometric mean directly, without calculating the common ratio. The stuff I wrote merely calculates the common ratio and then we use it to calculate each successive Geometric Mean.

Answer 8

What did you get?

Answer 9

12,36,108,324?

Answer 10

You have a common ratio of 3.
9724 / 324 = 30.012345679 -- Something Fishy, there.
324 * 3 = 972 -- Were we shooting for 972 or 9724?

Answer 11

I re read the problem and my teacher forgot to put a period. "Find the four geometric means between 4 and 9724,___,___,___,___,972Enter your answers separated by a comma."
So I was supposed to go from 4-972

Answer 12

ahahaha. That makes SO MUCH more sense. Good work!

Answer 13

Seriously haha, thank you so much!