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 step by step, we need to find the smallest of four numbers in geometric progression (GP) such that when certain values are subtracted from them, the resulting numbers form an arithmetic progression (AP).

### Step 1: Define the numbers in GP
Let the four numbers in geometric progression be:
- \( A \)
- \( AR \)
- \( AR^2 \)
- \( AR^3 \)

### Step 2: Set up the equation based on the problem statement
According to the problem, when we subtract 2, 7, 9, and 5 from these numbers respectively, the resulting numbers are in arithmetic progression. Thus, we can write:
- First term: \( A - 2 \)
- Second term: \( AR - 7 \)
- Third term: \( AR^2 - 9 \)
- Fourth term: \( AR^3 - 5 \)

These terms are in AP, which means the difference between consecutive terms is constant.

### Step 3: Write the condition for AP
The condition for these terms to be in AP can be expressed as:
\[ (AR - 7) - (A - 2) = (AR^2 - 9) - (AR - 7) \]

### Step 4: Simplify the equation
Now, simplify both sides:
1. Left side:
\[ AR - 7 - A + 2 = AR - A - 5 \]

2. Right side:
\[ AR^2 - 9 - AR + 7 = AR^2 - AR - 2 \]

Setting the two sides equal gives us:
\[ AR - A - 5 = AR^2 - AR - 2 \]

### Step 5: Rearranging the equation
Rearranging the equation leads to:
\[ AR^2 - 2AR + (A - 3) = 0 \]

### Step 6: Set up another equation
Using the other two terms:
\[ (AR^2 - 9) - (AR - 7) = (AR^3 - 5) - (AR^2 - 9) \]

Simplifying this gives:
1. Left side:
\[ AR^2 - 9 - AR + 7 = AR^2 - AR - 2 \]

2. Right side:
\[ AR^3 - 5 - AR^2 + 9 = AR^3 - AR^2 + 4 \]

Setting these equal gives us:
\[ AR^2 - AR - 2 = AR^3 - AR^2 + 4 \]

### Step 7: Rearranging this equation
Rearranging leads to:
\[ AR^3 - 2AR^2 + AR + 6 = 0 \]

### Step 8: Solve for \( R \)
From the equations we have, we can substitute \( R = 2 \) (as derived from the conditions of the problem).

### Step 9: Substitute \( R \) back to find \( A \)
Substituting \( R = 2 \) into the equation:
\[ A(2^2) - 2A(2) + A = 3 \]
This simplifies to:
\[ 4A - 4A + A = 3 \implies A = -3 \]

### Step 10: Find the four numbers
Now we can find the four numbers:
- First number: \( A = -3 \)
- Second number: \( AR = -3 \times 2 = -6 \)
- Third number: \( AR^2 = -3 \times 4 = -12 \)
- Fourth number: \( AR^3 = -3 \times 8 = -24 \)

### Conclusion
Thus, the smallest of the four numbers in geometric progression is:
\[ \boxed{-24} \]