III. Direction: Insert geometric means in each geometric sequence.
1. 3, _, 8
2. 32, _,38
3. 7, _, _, 56
4. 36, _, _, _, 964
5.-1, _, -9, _, -81

To find the geometric means in a sequence, we use the property of geometric sequences where each term is obtained by multiplying the previous term by a constant ratio (r).

### 1. Sequence: 3, _, 8

Let the missing term be \( x \). 

Given: 
- First term \( a_1 = 3 \)
- Third term \( a_3 = 8 \)

The sequence formula for the third term in a geometric sequence is:
\[
a_3 = a_1 \cdot r^2
\]

Plugging in the known values:
\[
8 = 3 \cdot r^2
\]

Solving for \( r^2 \):
\[
r^2 = \frac{8}{3}
\]

\[
r = \sqrt{\frac{8}{3}} \approx 1.632
\]

Now, the second term \( a_2 \) can be found as:
\[
a_2 = 3 \cdot r = 3 \cdot \sqrt{\frac{8}{3}} \approx 4.9
\]

### 2. Sequence: 32, _, 38

Let the missing term be \( x \).

Given:
- First term \( a_1 = 32 \)
- Third term \( a_3 = 38 \)

Sequence formula:
\[
a_3 = a_1 \cdot r^2
\]

Plugging in:
\[
38 = 32 \cdot r^2
\]

Solving for \( r^2 \):
\[
r^2 = \frac{38}{32}
\]

\[
r = \sqrt{\frac{38}{32}} \approx 1.1
\]

So the missing term:
\[
a_2 = 32 \cdot r = 32 \times 1.1 \approx 35.2
\]

### 3. Sequence: 7, _, _, 56

Let the missing terms be \( a_2 \) and \( a_3 \).

Given:
- First term \( a_1 = 7 \)
- Fourth term \( a_4 = 56 \)

\[
a_4 = a_1 \cdot r^3
\]

Plugging in values:
\[
56 = 7 \cdot r^3
\]

Solving for \( r^3 \):
\[
r^3 = \frac{56}{7} = 8
\]

\[
r = \sqrt[3]{8} = 2
\]

So:
\[
a_2 = 7 \cdot 2 = 14, \quad a_3 = 14 \cdot 2 = 28
\]

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

Let the missing terms be \( a_2 \), \( a_3 \), and \( a_4 \).

Given:
- First term \( a_1 = 36 \)
- Fifth term \( a_5 = 964 \)

\[
a_5 = a_1 \cdot r^4
\]

Plugging in:
\[
964 = 36 \cdot r^4
\]

Solving for \( r^4 \):
\[
r^4 = \frac{964}{36}
\]

\[
r = \sqrt[4]{\frac{964}{36}} \approx 3.1
\]

Thus:
\[
a_2 = 36 \times 3.1 = 112 \approx, a_3 = 112 \times 3.1 = 348, a_4 = 348 \times 3.1 = 964
\]

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

Let missing terms be \( a_2 \) and \( a_4 \).

Given:
- \( a_1 = -1 \)
- \( a_3 = -9 \)
- \( a_5 = -81 \)

Formula:
\[
r^2 = a_1 = \frac{-81/-1} \approx = 1.1
\]

Simplifying:
a_4 = \( -9 \)



Explore how to find missing terms using geometric means in geometric sequences with detailed examples and solutions. Understand the properties of geometric sequences and learn to apply the formula for the nth term. Suitable for high school students studying sequences and series.

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