If a, b, c are three unequal numbers such that a, b, c are in arithmetic progression and $b-a$, $c-b$, a are in geometric progression, then $a:b:c$ is:

Question

Accepted Answer

To solve the problem, we need to analyze the conditions given for the numbers $a$, $b$, and $c$.

1. **Arithmetic Progression Condition**: The numbers $a$, $b$, and $c$ are in arithmetic progression. This means:
   $$
   b = a + d \quad \text{and} \quad c = a + 2d
   $$
   where $d$ is the common difference.

2. **Geometric Progression Condition**: The numbers $b-a$, $c-b$, and $a$ are in geometric progression. This implies:
   $$
   b-a = d, \quad c-b = d, \quad \text{and} \quad a
   $$
   are in geometric progression. Therefore, the ratio between consecutive terms is constant:
   $$
   \frac{c-b}{b-a} = \frac{a}{c-b}
   $$
   Substituting the values, we get:
   $$
   \frac{d}{d} = \frac{a}{d}
   $$
   Simplifying gives:
   $$
   1 = \frac{a}{d}
   $$
   Therefore, $a = d$.

Given these conditions, we can express $b$ and $c$ in terms of $a$:
- Since $a = d$, we have $b = a + a = 2a$ and $c = a + 2a = 3a$.

Thus, the ratio $a:b:c$ is:
$$
a:b:c = a:2a:3a = 1:2:3
$$

Now, let's match this ratio with the given options. The correct answer is Option 3, which corresponds to the ratio $1:2:3$ when interpreted correctly in the context of the problem.

### Explanation of Options:

- **Option 1**: This option does not correspond to the ratio $1:2:3$. It is incorrect because it does not match the derived ratio from the conditions given in the problem.
  
- **Option 2**: Similar to Option 1, this option does not represent the ratio $1:2:3$. It is incorrect for the same reason.

- **Option 3**: This option is correct as it corresponds to the ratio $1:2:3$, which is derived from the conditions of the problem.

- **Option 4**: This option also does not match the ratio $1:2:3$ and is therefore incorrect.

In conclusion, Option 3 is the correct answer because it accurately reflects the ratio $a:b:c = 1:2:3$ derived from the given conditions of arithmetic and geometric progressions.