If 2, 7, 9 and 5 are subtraced respectively from four numbers in geometric progression, then the resulting numbers are in arithmetic progression. The smallest of the four numbers is

Options

A. `-24`
B. `-12`
C. 6
D. 3

Answer

To solve the problem, we need to determine the smallest of four numbers in geometric progression (GP) such that when specific values are subtracted from them, the resulting numbers form an arithmetic progression (AP).

### Step-by-Step Solution:

1. **Define the Numbers in GP**:
Let the four numbers in GP be:
- First term: \( A \)
- Second term: \( AR \)
- Third term: \( AR^2 \)
- Fourth term: \( AR^3 \)

2. **Subtract the Given Values**:
According to the problem, we subtract the following values from the GP numbers:
- From \( A \), subtract 2: \( A - 2 \)
- From \( AR \), subtract 7: \( AR - 7 \)
- From \( AR^2 \), subtract 9: \( AR^2 - 9 \)
- From \( AR^3 \), subtract 5: \( AR^3 - 5 \)

3. **Set Up the AP Condition**:
The resulting numbers must be in AP. For numbers to be in AP, the difference between consecutive terms must be constant. Therefore, we set up the equations based on the differences:
\[ (AR - 7) - (A - 2) = (AR^2 - 9) - (AR - 7) \]
\[ (AR^2 - 9) - (AR - 7) = (AR^3 - 5) - (AR^2 - 9) \]

4. **Simplify the First Equation**:
Expanding the first equation:
\[ AR - 7 - A + 2 = AR^2 - 9 - AR + 7 \]
Simplifying gives:
\[ AR - A - 5 = AR^2 - AR - 2 \]
Rearranging, we get:
\[ AR^2 - 2AR + A = -3 \quad \text{(Equation 1)} \]

5. **Simplify the Second Equation**:
Expanding the second equation:
\[ AR^2 - 9 - AR + 7 = AR^3 - 5 - AR^2 + 9 \]
Simplifying gives:
\[ AR^2 - AR - 2 = AR^3 - 5 - AR^2 + 9 \]
Rearranging, we get:
\[ AR^3 - 2AR^2 + AR = -6 \quad \text{(Equation 2)} \]

6. **Substituting Equation 1 into Equation 2**:
From Equation 1, we can express \( A \) in terms of \( R \):
\[ A = -3 + 2AR - AR^2 \]
Substitute this into Equation 2 and solve for \( R \).

7. **Solve for \( R \)**:
After substituting and simplifying, we find that \( R = 2 \).

8. **Find \( A \)**:
Substitute \( R = 2 \) back into Equation 1 to find \( A \):
\[ A = -3 + 2A(2) - (2^2)A \]
This simplifies to \( A = -3 \).

9. **Calculate the Four Numbers**:
Now we can calculate the four numbers:
- First term: \( A = -3 \)
- Second term: \( AR = -3 \times 2 = -6 \)
- Third term: \( AR^2 = -3 \times 2^2 = -12 \)
- Fourth term: \( AR^3 = -3 \times 2^3 = -24 \)

10. **Determine the Smallest Number**:
The four numbers are \( -3, -6, -12, -24 \). The smallest of these numbers is \( -24 \).

### Conclusion:
The smallest of the four numbers in geometric progression is \( \boxed{-24} \).