To find the geometric means in a sequence, we first need to recognize that in a geometric sequence, the ratio between consecutive terms is constant. This constant ratio is called the "common ratio" and is denoted by $r$.

Given the first and last terms of the sequence, we can calculate this ratio and then use it to fill in the missing geometric means.

Let's solve each problem step by step.

1. Sequence: 3, _, 8

Let the missing term be $x$.

• The common ratio $r$ is the same between all terms.
• $8 = 3 \cdot r^2$.

To find $r$: $r^2 = \frac{8}{3} \quad \Rightarrow \quad r = \sqrt{\frac{8}{3}}.$

Now find $x = 3 \cdot r$: $x = 3 \cdot \sqrt{\frac{8}{3}}.$

2. Sequence: 32, _, 38

This sequence is not a geometric sequence because $38$ is not a multiple of $32$. Are you sure this is supposed to be a geometric sequence?

3. Sequence: 7, _, _, 56

Here, there are two missing terms. Let the common ratio be $r$.

• $56 = 7 \cdot r^3$.

To find $r$: $r^3 = \frac{56}{7} = 8 \quad \Rightarrow \quad r = 2.$ Now, calculate the missing terms:

• First term: $7 \cdot r = 7 \cdot 2 = 14$,
• Second term: $14 \cdot r = 14 \cdot 2 = 28$.

So, the sequence is $7, 14, 28, 56$.

4. Sequence: 36, _, _, _, 964

Here, there are three missing terms. Let the common ratio be $r$.

• $964 = 36 \cdot r^4$.

To find $r$: $r^4 = \frac{964}{36} \quad \Rightarrow \quad r = \sqrt[4]{\frac{964}{36}}.$

Then, calculate the missing terms using this ratio.

5. Sequence: -1, _, -9, _, -81

There are two missing terms. Let the common ratio be $r$.

• $-9 = -1 \cdot r^2$.

To find $r$: $r^2 = 9 \quad \Rightarrow \quad r = \pm 3.$ Now, calculate the missing terms using both possible values for $r$.

Would you like to continue with the detailed calculations, or do you have any questions so far? Here are some related questions to explore further:

1. How do you calculate the common ratio in a geometric sequence?
2. What happens if the sequence isn't geometric? How do you check?
3. How do you handle negative numbers in geometric sequences?
4. What are the properties of geometric sequences?
5. How can you calculate the sum of a geometric sequence?

Tip: Always check if a sequence is truly geometric by confirming the common ratio between all consecutive terms.