The 5th term in a geometric sequence is 120. The 7th term is 30. What are the possible values of the 6th term of the sequence?

Answer 1

Let's call the first term of the sequence "a" and the common ratio "r". We can use the formula for the nth term of a geometric sequence to set up two equations:

a * r^4 = 120 (since the 5th term is a * r^4)
a * r^6 = 30 (since the 7th term is a * r^6)

Now we can solve for "a" and "r". One way to do this is to divide the second equation by the first:

(r^6)/(r^4) = 30/120

Simplifying, we get:

r^2 = 1/4

Taking the square root of both sides, we get:

r = +/- 1/2

Now we can substitute these values of "r" back into either equation to solve for "a". Let's use the first equation:

a * (1/2)^4 = 120

Simplifying, we get:

a = 960

So our possible sequences are:

960, 480, 240, 120, 60, 30, ...

or

960, -480, 240, -120, 60, -30, ...

Therefore, the possible values of the 6th term of the sequence are 60 and -30.

Answer 2

are you sure

Answer 3

I apologize for the mistake in my previous response. The correct solution is as follows:

Using the same approach as before, we can set up the equations:

a * r^4 = 120
a * r^6 = 30

Dividing the second equation by the first, we get:

r^2 = 1/4

Taking the square root, we get:

r = +/- 1/2

Substituting each value of "r" into the first equation to solve for "a", we get:

When r = 1/2:

a = 120 / (1/2)^4 = 1920

When r = -1/2:

a = 120 / (-1/2)^4 = -1920

Therefore, the two possible sequences are:

1920, 960, 480, 240, 120, 60, ...
and
-1920, -960, -480, -240, -120, -60, ...

And the possible values of the 6th term are 60 and -60. I apologize for any confusion caused by my previous response.

Answer 4

is this correct

Answer 5

Yes, that is correct. I apologize for any mistakes in my previous responses.