If a,b,c are in A.P. and `a^2,b^2,c^2` are in H.P. then which of the following could and true (A) `-a/2, b, c are in G.P.` (B) `a=b=c` (C) `a^3,b^3,c^3` are in G.P. (D) none of these

Options

A. `-(a)/(2),b,c` are in GP
B. `a=b=c`
C. `a^(2),b^(2),c^(2)` are in GP
D. None of these

Explanation

To solve the problem step-by-step, we need to analyze the conditions given: \(a, b, c\) are in Arithmetic Progression (A.P.) and \(a^2, b^2, c^2\) are in Harmonic Progression (H.P.).

### Step 1: Understanding A.P. and H.P.
1. Since \(a, b, c\) are in A.P., we have:
\[ b = \frac{a + c}{2} \]
This is our first equation.

2. Since \(a^2, b^2, c^2\) are in H.P., we can express this condition as:
\[ \frac{1}{b^2} = \frac{2}{a^2 + c^2} \]
This is our second equation.

### Step 2: Substitute and Equate
3. From the first equation, substitute \(b\) into the second equation:
\[ b^2 = \left(\frac{a + c}{2}\right)^2 = \frac{(a + c)^2}{4} \]
Now substituting this into the H.P. condition gives:
\[ \frac{4}{(a + c)^2} = \frac{2}{a^2 + c^2} \]

### Step 3: Cross Multiplying
4. Cross-multiply to eliminate the fractions:
\[ 4(a^2 + c^2) = 2(a + c)^2 \]
Expanding the right side:
\[ 4(a^2 + c^2) = 2(a^2 + 2ac + c^2) \]
This simplifies to:
\[ 4a^2 + 4c^2 = 2a^2 + 4ac + 2c^2 \]

### Step 4: Rearranging the Equation
5. Rearranging gives:
\[ 4a^2 + 4c^2 - 2a^2 - 4ac - 2c^2 = 0 \]
Simplifying this yields:
\[ 2a^2 + 2c^2 - 4ac = 0 \]
Dividing through by 2:
\[ a^2 + c^2 - 2ac = 0 \]

### Step 5: Factoring
6. This can be factored as:
\[ (a - c)^2 = 0 \]
Thus, we find:
\[ a = c \]

### Step 6: Finding \(b\)
7. Substitute \(a = c\) back into the equation for \(b\):
\[ b = \frac{a + c}{2} = \frac{a + a}{2} = a \]
Therefore, we conclude:
\[ a = b = c \]

### Step 7: Analyzing the Options
8. Now we analyze the options given in the question:
- (A) \(-\frac{a}{2}, b, c\) are in G.P.:
- If \(a = b = c\), then \(-\frac{a}{2}, a, a\) are not in G.P.
- (B) \(a = b = c\):
- This is true.
- (C) \(a^3, b^3, c^3\) are in G.P.:
- If \(a = b = c\), then \(a^3, a^3, a^3\) are in G.P.
- (D) None of these:
- This is not true since (B) and (C) are true.

### Conclusion
The correct answers are (B) and (C).