If a,b, and c are in A.P and b-a,c-b and a are in G.P then a:b:c is

Options

A. `2:3:4`
B. `1:2:4`
C. `1:2:3`
D. `1:3:5`

Answer

To solve the problem, we need to find the ratio \( a:b:c \) given that \( a, b, c \) are in Arithmetic Progression (A.P) and \( b-a, c-b, a \) are in Geometric Progression (G.P).

### Step-by-Step Solution:

1. **Understanding A.P**:
Since \( a, b, c \) are in A.P, we can express \( b \) and \( c \) in terms of \( a \):
\[ b = a + d \quad \text{and} \quad c = a + 2d \]
where \( d \) is the common difference.

2. **Finding the differences**:
The differences \( b-a \) and \( c-b \) can be calculated as:
\[ b - a = d \quad \text{and} \quad c - b = (a + 2d) - (a + d) = d \]

3. **Setting up the G.P condition**:
We are given that \( b-a, c-b, a \) are in G.P. This means:
\[ \frac{c-b}{b-a} = \frac{a}{c-b} \]
Substituting the values we found:
\[ \frac{d}{d} = \frac{a}{d} \]
This simplifies to:
\[ 1 = \frac{a}{d} \]
Therefore, we have:
\[ a = d \]

4. **Substituting back**:
Now substituting \( d \) back into the expressions for \( b \) and \( c \):
\[ b = a + d = a + a = 2a \]
\[ c = a + 2d = a + 2a = 3a \]

5. **Finding the ratio**:
Now we have:
\[ a = a, \quad b = 2a, \quad c = 3a \]
Thus, the ratio \( a:b:c \) is:
\[ a:b:c = a:2a:3a = 1:2:3 \]

### Final Answer:
The ratio \( a:b:c \) is \( 1:2:3 \).

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