Find a three digit number if its digits form a geometric progression and the digits of the number which is
smallar by 400 form an A.P. is

Options

A. `139`
B. `248`
C. `842`
D. `931`

Answer

To solve the problem of finding a three-digit number whose digits form a geometric progression (G.P.) and the digits of the number which is smaller by 400 form an arithmetic progression (A.P.), we can follow these steps:

### Step 1: Understanding the Problem
We need to find a three-digit number represented as \(abc\), where \(a\), \(b\), and \(c\) are its digits. The conditions are:
1. The digits \(a\), \(b\), and \(c\) must be in G.P.
2. The number \(abc - 400\) must have its digits in A.P.

### Step 2: Formulating the G.P. Condition
For the digits \(a\), \(b\), and \(c\) to be in G.P., the following condition must hold:
\[ b^2 = a \cdot c \]

### Step 3: Formulating the A.P. Condition
If we denote the three-digit number as \(N = 100a + 10b + c\), then we need to check:
\[ N - 400 = 100a + 10b + c - 400 \]
Let \(M = N - 400\). The digits of \(M\) must be in A.P. If \(M\) has digits \(x\), \(y\), and \(z\), then the condition for A.P. is:
\[ 2y = x + z \]

### Step 4: Testing Possible Three-Digit Numbers
We can test three-digit numbers systematically. We will check the digits of the number and see if they satisfy both conditions.

#### Option 1: Testing 842
1. Check if the digits \(8\), \(4\), and \(2\) form a G.P.:
\[ 4^2 = 8 \cdot 2 \Rightarrow 16 = 16 \quad \text{(True)} \]
2. Calculate \(842 - 400 = 442\) and check if \(4\), \(4\), and \(2\) form an A.P.:
\[ 2 \cdot 4 = 4 + 2 \Rightarrow 8 = 6 \quad \text{(False)} \]

#### Option 2: Testing 931
1. Check if the digits \(9\), \(3\), and \(1\) form a G.P.:
\[ 3^2 = 9 \cdot 1 \Rightarrow 9 = 9 \quad \text{(True)} \]
2. Calculate \(931 - 400 = 531\) and check if \(5\), \(3\), and \(1\) form an A.P.:
\[ 2 \cdot 3 = 5 + 1 \Rightarrow 6 = 6 \quad \text{(True)} \]

### Conclusion
Since both conditions are satisfied for the number 931, we conclude that the three-digit number we are looking for is:
\[ \boxed{931} \]