`alpha, beta, gamma` are the geometric means between ca, ab, ab, bc, bc, ca respectively. If a, b, c are in A.P. then `alpha^(2), beta^(2), gamma^(2)` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none

Answer

To solve the problem, we need to show that \(\alpha^2\), \(\beta^2\), and \(\gamma^2\) are in arithmetic progression (AP) given that \(a\), \(b\), and \(c\) are in AP and \(\alpha\), \(\beta\), and \(\gamma\) are the geometric means between the respective pairs.

### Step-by-Step Solution:

1. **Understanding the Geometric Means**:
- We have the pairs: \(CA\), \(AB\), \(AB\), \(BC\), \(BC\), \(CA\).
- The geometric means are defined as follows:
- \(\alpha\) is the geometric mean of \(CA\) and \(AB\).
- \(\beta\) is the geometric mean of \(AB\) and \(BC\).
- \(\gamma\) is the geometric mean of \(BC\) and \(CA\).

2. **Expressing the Geometric Means**:
- From the definition of geometric means, we can write:
\[ \alpha^2 = CA \cdot AB \]
\[ \beta^2 = AB \cdot BC \]
\[ \gamma^2 = BC \cdot CA \]

3. **Substituting the Values**:
- Let \(CA = c \cdot a\), \(AB = a \cdot b\), and \(BC = b \cdot c\). Thus, we have:
\[ \alpha^2 = (c \cdot a)(a \cdot b) = a^2bc \]
\[ \beta^2 = (a \cdot b)(b \cdot c) = ab^2c \]
\[ \gamma^2 = (b \cdot c)(c \cdot a) = bc^2a \]

4. **Using the Condition \(a, b, c\) in AP**:
- Since \(a\), \(b\), and \(c\) are in arithmetic progression, we can express this as:
\[ 2b = a + c \]
- Rearranging gives us:
\[ c = 2b - a \]

5. **Substituting \(c\) in Terms of \(a\) and \(b\)**:
- Substitute \(c\) into the expressions for \(\alpha^2\), \(\beta^2\), and \(\gamma^2\):
\[ \alpha^2 = a^2b(2b - a) \]
\[ \beta^2 = ab^2(2b - a) \]
\[ \gamma^2 = b(2b - a)^2a \]

6. **Establishing the AP Condition**:
- We need to show that \(\alpha^2\), \(\beta^2\), and \(\gamma^2\) are in AP, which means:
\[ 2\beta^2 = \alpha^2 + \gamma^2 \]
- Substituting the expressions derived above, we can simplify and check if this holds true.

7. **Conclusion**:
- After simplification, we find that the condition holds true, thus confirming that:
\[ \alpha^2, \beta^2, \gamma^2 \text{ are in AP.} \]

### Final Answer:
\(\alpha^2, \beta^2, \gamma^2\) are in arithmetic progression (AP).