If three non-zero distinct real numbers form an arithmatic progression and the squares of these numbers taken in the same order constitute a geometric progression. Find the sum of all possible common ratios of the geometric progression.

Answer

To solve the problem, we need to find three distinct non-zero real numbers that form an arithmetic progression (AP) and whose squares form a geometric progression (GP). Let's denote the three numbers in the AP as \( A - D \), \( A \), and \( A + D \).

### Step-by-step Solution:

1. **Define the Numbers in AP**:
Let the three numbers be:
\[ x_1 = A - D, \quad x_2 = A, \quad x_3 = A + D \]
These numbers are in arithmetic progression.

2. **Square the Numbers**:
The squares of these numbers are:
\[ x_1^2 = (A - D)^2, \quad x_2^2 = A^2, \quad x_3^2 = (A + D)^2 \]

3. **Condition for GP**:
The squares of these numbers must satisfy the condition for a geometric progression:
\[ x_2^2 = \sqrt{x_1^2 \cdot x_3^2} \]
This can be rewritten as:
\[ A^2 = \sqrt{(A - D)^2 \cdot (A + D)^2} \]

4. **Simplify the Condition**:
Squaring both sides gives:
\[ A^4 = (A - D)^2 \cdot (A + D)^2 \]
Expanding the right-hand side:
\[ A^4 = ((A - D)(A + D))^2 = (A^2 - D^2)^2 \]
Therefore, we have:
\[ A^4 = A^4 - 2A^2D^2 + D^4 \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ 0 = -2A^2D^2 + D^4 \]
This can be factored as:
\[ D^2(D^2 - 2A^2) = 0 \]

6. **Solving the Factors**:
We have two cases:
- \( D^2 = 0 \) (not possible since \( D \) must be non-zero)
- \( D^2 - 2A^2 = 0 \) which gives \( D^2 = 2A^2 \) or \( D = \pm \sqrt{2}A \)

7. **Finding the Common Ratio**:
The common ratio \( R \) of the GP can be calculated as:
\[ R = \frac{x_2^2}{x_1^2} = \frac{A^2}{(A - D)^2} \]
Substituting \( D = \sqrt{2}A \):
\[ R = \frac{A^2}{(A - \sqrt{2}A)^2} = \frac{A^2}{(1 - \sqrt{2})^2 A^2} = \frac{1}{(1 - \sqrt{2})^2} \]

Similarly, substituting \( D = -\sqrt{2}A \):
\[ R = \frac{A^2}{(A + \sqrt{2}A)^2} = \frac{A^2}{(1 + \sqrt{2})^2 A^2} = \frac{1}{(1 + \sqrt{2})^2} \]

8. **Calculating the Values of R**:
Now we need to compute the values:
\[ R_1 = \frac{1}{(1 - \sqrt{2})^2}, \quad R_2 = \frac{1}{(1 + \sqrt{2})^2} \]

9. **Finding the Sum of Common Ratios**:
The sum of all possible common ratios is:
\[ R_1 + R_2 = \frac{1}{(1 - \sqrt{2})^2} + \frac{1}{(1 + \sqrt{2})^2} \]

To simplify this, we can use the identity:
\[ (a - b)(a + b) = a^2 - b^2 \]
Thus, we can find a common denominator and simplify.

10. **Final Calculation**:
After simplification, we find that:
\[ R_1 + R_2 = 6 \]

### Conclusion:
The sum of all possible common ratios of the geometric progression is:
\[ \boxed{6} \]