To solve the problem of finding the correct relationship between \( x \) and \( y \), where \( x \) and \( y \) are the two geometric means (G.M's) between 4 and 108, we follow a step-by-step approach:
1. Given the first term \( a_1 = 4 \) and the third term \( a_3 = 108 \).
2. We need to find the two geometric means \( x \) and \( y \) such that these means form a geometric sequence.
3. In a geometric sequence, the \( n \)-th term can be expressed as \( a_n = a_1 \cdot r^{(n-1)} \). Hence:
[tex]\[
a_2 = a_1 \cdot r \quad \text{and} \quad a_3 = a_1 \cdot r^2
\][/tex]
4. We have:
[tex]\[
108 = 4 \cdot r^2
\][/tex]
[tex]\[
r^2 = \frac{108}{4} = 27
\][/tex]
[tex]\[
r = \sqrt{27} \approx 5.196
\][/tex]
5. Using \( r \), we can determine \( x \) and \( y \):
[tex]\[
x = a_1 \cdot r = 4 \cdot 5.196 \approx 20.785
\][/tex]
[tex]\[
y = a_1 \cdot r^2 = a_1 \cdot 27 = 4 \cdot 27 = 108
\][/tex]
6. Now, we check the given options for relationships between \( x \) and \( y \):
- a) \( x = y \):
[tex]\[
20.785 \neq 108
\][/tex]
This option is False.
- b) \( y = 3x \):
[tex]\[
108 = 3 \times 20.785 \approx 62.355
\][/tex]
This option is False.
- c) \( x = 2y \):
[tex]\[
20.785 = 2 \times 108 = 216
\][/tex]
This option is False.
- d) \( y = 2x \):
[tex]\[
108 = 2 \times 20.785 \approx 41.570
\][/tex]
This option is False.
Given the calculations, none of the provided options (a, b, c, or d) are correct. Therefore, there is no correct match from the given choices.
