Question

Find the geometric means in the following sequence 30,?,?,?,?,-3900000

Answer 1

is that \(-3,900,000\) at then end or \(390,000\) ?

Answer 2

-3900000

Answer 3

\[30, 30r,30r^2,30r^3,30r^4,30r^5=-3,900,000\]
\[\frac{-3,900,000}{30}=r^5\]
\[r^5=-130,000\]
\[r=\sqrt[5]{-130,000}\]

Answer 4

looks kind of ugly, did i write something incorrectly?

Answer 5

maybe i am completely wrong.

Answer 6

Looks right to me. the common ratio is between -11 and -10.

Answer 7

\[r=-10^{\frac{4}{5}}\sqrt[5]{13}\]hmmm

Answer 8

So it would be 390,3900,39000,390000

Answer 9

Does the answer need to be a simplified expression of an exact value or will a decimal approximation do?

Answer 10

Am I right?

Answer 11

Are you sure? Definition is:
\[\text{If } (x_1,\dots,x_n) \text{ then } G=\left(x_1\cdots x_n\right)^{\frac{1}{n}}\]

Answer 12

good point @Limitless

Answer 13

Wiki: "For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product; that is 2√2 × 8 = 4. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2; that is 3√4 × 1 × 1/32 = ½."

Answer 14

i wrote the common ratio, not the geometric mean

Answer 15

I don't know what the ?'s are in:
30,?,?,?,?,-3900000

The GM is not defined unless we know them...

Answer 16

If the common ratio is \[\sqrt[5]{\frac{-3900000}{30}}\]
Then the terms are (approximately) 30, -316.16, 3331.95, -35114.56, 370063.23, -3900000

Answer 17

I think by "Find the geometric means," it means to find the missing terms.

Answer 18

i was assuming it was a geometric sequence,
\[\{a,ar, ar^2, ar^3, ar^4, ar^5\}\] but i could be wrong
in which case you would have geometric mean as
\[\sqrt[6]{a^6r^{15}}=a\sqrt[3]{r^5}\]

Answer 19

OP needs to clarify. We can conjecture a lot and get nowhere. :P

Answer 20

this looks messy too, but who knows

Answer 21

I think Satellite has it: \[GM=30x^{5/3}, x=\sqrt[5]{\frac{-3900000}{30}}\]
\[\rightarrow 30\sqrt[3]{-130000}\]