If a, b, c are in continued proportion, then which of the following to follow.

Options

A. `b^2 = ac`
B. `c` is the third proportional
C. `a/b = c/b`
D. `a/b = d/c`

Answer

To solve the problem, we need to determine the relationships that hold true when three quantities \( a, b, c \) are in continued proportion.

### Step-by-Step Solution:

1. **Understanding Continued Proportion**:
- When three quantities \( a, b, c \) are said to be in continued proportion, it means that the ratio of the first quantity to the second is equal to the ratio of the second quantity to the third.
- Mathematically, this is expressed as:
\[ \frac{a}{b} = \frac{b}{c} \]

2. **Cross Multiplying**:
- From the equation \( \frac{a}{b} = \frac{b}{c} \), we can cross-multiply to derive a relationship between \( a, b, \) and \( c \):
\[ a \cdot c = b \cdot b \]
- This simplifies to:
\[ ac = b^2 \]

3. **Rearranging**:
- We can rearrange the equation to highlight \( b^2 \):
\[ b^2 = ac \]

4. **Identifying the Third Proportional**:
- In the context of continued proportion, the quantity \( c \) can be referred to as the third proportional to \( a \) and \( b \). This means that if \( a \) and \( b \) are the first two terms, then \( c \) is the third term in the proportion.

5. **Analyzing the Options**:
- Now, let’s evaluate the options provided:
- **Option 1**: \( b^2 = ac \) (This is correct)
- **Option 2**: \( c \) is the third proportional (This is also correct)
- **Option 3**: \( \frac{a}{b} = \frac{c}{b} \) (This is not necessarily true)
- **Option 4**: \( \frac{a}{b} = \frac{d}{c} \) (This is not relevant to the given quantities)

6. **Conclusion**:
- The correct options are **Option 1** and **Option 2**.