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

Explanation

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 this relationship as:
\[ b = \frac{a + c}{2} \]

2. **Understanding G.P.**:
The terms \( b-a, c-b, a \) are in G.P. This means that:
\[ (c-b)^2 = (b-a) \cdot a \]

3. **Substituting for \( b \)**:
Substitute \( b \) from the A.P. relationship into the G.P. equation:
\[ c - b = c - \frac{a+c}{2} = \frac{2c - a - c}{2} = \frac{c - a}{2} \]
\[ b - a = \frac{a+c}{2} - a = \frac{c - a}{2} \]

4. **Setting up the G.P. equation**:
Now substituting these into the G.P. condition:
\[ \left(\frac{c - a}{2}\right)^2 = \left(\frac{c - a}{2}\right) \cdot a \]

5. **Simplifying the equation**:
This simplifies to:
\[ \frac{(c - a)^2}{4} = \frac{(c - a) \cdot a}{2} \]
Multiplying both sides by 4 to eliminate the fraction:
\[ (c - a)^2 = 2a(c - a) \]
Rearranging gives:
\[ (c - a)^2 - 2a(c - a) = 0 \]
Factoring out \( (c - a) \):
\[ (c - a)((c - a) - 2a) = 0 \]

6. **Finding values**:
This gives us two cases:
- Case 1: \( c - a = 0 \) which implies \( c = a \) (not valid since \( a, b, c \) must be distinct).
- Case 2: \( c - a = 2a \) which implies \( c = 3a \).

7. **Finding \( b \)**:
Now substituting \( c = 3a \) back into the A.P. equation:
\[ b = \frac{a + c}{2} = \frac{a + 3a}{2} = \frac{4a}{2} = 2a \]

8. **Final Ratios**:
Now we have:
\[ a = a, \quad b = 2a, \quad c = 3a \]
Therefore, the ratio \( a:b:c \) is:
\[ a:b:c = a:2a:3a = 1:2:3 \]

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