Answer :
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.
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.
