Let `a,b,c`, are non-zero real numbers such that `(1)/(a),(1)/(b),(1)/(c )` are in arithmetic progression and `a,b,-2c`, are in geometric progression, then which of the following statements (s) is (are) must be true?

Options

A. `a^(2),b^(2), 4c^(2)` are in geometric progression
B. `-2a,b, -2c` are in arithmetic progression
C. `a^(3)+b^(3)+c^(3) - 3abc = 0`
D. `a^(2),b^(2),c^(2)` are in harmonic progression

Explanation

To solve the problem, we need to analyze the given conditions step by step.

### Step 1: Understand the conditions
We are given two conditions:
1. \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in arithmetic progression (AP).
2. \( a, b, -2c \) are in geometric progression (GP).

### Step 2: Use the condition of AP
For three numbers \( x, y, z \) to be in AP, the condition is:
\[ 2y = x + z \]

Applying this to our case:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]

Multiplying through by \( abc \) to eliminate the denominators gives:
\[ 2ac = bc + ab \]
Rearranging this, we get:
\[ ab + bc - 2ac = 0 \]
This is our first equation (Equation 1).

### Step 3: Use the condition of GP
For three numbers \( x, y, z \) to be in GP, the condition is:
\[ y^2 = xz \]

Applying this to our case:
\[ b^2 = a \cdot (-2c) \]
This simplifies to:
\[ b^2 = -2ac \]
This is our second equation (Equation 2).

### Step 4: Substitute and solve
From Equation 2, we have:
\[ ac = -\frac{b^2}{2} \]

Substituting this into Equation 1:
\[ ab + bc - 2\left(-\frac{b^2}{2}\right) = 0 \]
This simplifies to:
\[ ab + bc + b^2 = 0 \]

### Step 5: Factor the equation
We can factor this equation:
\[ b(a + b + c) = 0 \]

Since \( b \) is a non-zero real number, we must have:
\[ a + b + c = 0 \]

### Step 6: Conclusion
Now, we can conclude that:
- The condition \( a + b + c = 0 \) must hold true.

### Step 7: Check for \( a^3 + b^3 + c^3 - 3abc = 0 \)
Using the identity for the sum of cubes, we have:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

Since \( a + b + c = 0 \), it follows that:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]

### Final Answer
Thus, the statement that must be true is:
\[ a^3 + b^3 + c^3 - 3abc = 0 \]