If a:b: :b:c then a,b,c are in _______

Answer

To solve the question "If a:b :: b:c then a, b, c are in _______", we will analyze the given ratios step by step.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
We are given that \( a:b \) is proportional to \( b:c \). This can be written as:
\[ \frac{a}{b} = \frac{b}{c} \]

2. **Cross Multiplying**:
To eliminate the fractions, we can cross-multiply:
\[ a \cdot c = b \cdot b \]
This simplifies to:
\[ ac = b^2 \]

3. **Identifying the Relationship**:
The equation \( ac = b^2 \) indicates that \( b \) is the geometric mean of \( a \) and \( c \). This means that \( a \), \( b \), and \( c \) are in a special relationship.

4. **Conclusion**:
Since \( b \) is the geometric mean of \( a \) and \( c \), we can conclude that \( a \), \( b \), and \( c \) are in **continued proportion**.

### Final Answer:
Thus, the answer is: **continued proportion**.

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