Let a, b, c be in AP.
Consider the following statements:
1. `(1)/(ab),(1)/(ca)" and "(1)/(bc)" are in AP."`
2. `(1)/(sqrt(b)+sqrt(c)),(1)/(sqrt(c)+sqrt(a))" and "(1)/(sqrt(a)+sqrt(b))" are in AP."`
Which of the statements given above is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To solve the problem, we need to verify whether the given statements about the sequences are correct.

### Step-by-Step Solution:

**Step 1: Understanding the Given Information**
- We are given that \( a, b, c \) are in Arithmetic Progression (AP). This means that:
\[ b - a = c - b \quad \text{or} \quad 2b = a + c \]

**Step 2: Analyzing the First Statement**
- The first statement claims that \( \frac{1}{ab}, \frac{1}{bc}, \frac{1}{ca} \) are in AP.
- For three numbers \( x, y, z \) to be in AP, the condition is:
\[ 2y = x + z \]
- Here, let:
\[ x = \frac{1}{ab}, \quad y = \frac{1}{bc}, \quad z = \frac{1}{ca} \]
- We need to check if:
\[ 2 \cdot \frac{1}{bc} = \frac{1}{ab} + \frac{1}{ca} \]

**Step 3: Simplifying the Equation**
- Multiply through by \( abc \) to eliminate the denominators:
\[ 2a = c + b \]
- Rearranging gives:
\[ 2b = a + c \]
- This is true since \( a, b, c \) are in AP. Thus, the first statement is **true**.

**Step 4: Analyzing the Second Statement**
- The second statement claims that \( \frac{1}{\sqrt{b} + \sqrt{c}}, \frac{1}{\sqrt{c} + \sqrt{a}}, \frac{1}{\sqrt{a} + \sqrt{b}} \) are in AP.
- Let:
\[ x = \frac{1}{\sqrt{b} + \sqrt{c}}, \quad y = \frac{1}{\sqrt{c} + \sqrt{a}}, \quad z = \frac{1}{\sqrt{a} + \sqrt{b}} \]
- We need to check if:
\[ 2y = x + z \]

**Step 5: Simplifying the Second Statement**
- Multiply through by \( (\sqrt{b} + \sqrt{c})(\sqrt{c} + \sqrt{a})(\sqrt{a} + \sqrt{b}) \) to eliminate the denominators:
\[ 2(\sqrt{c} + \sqrt{a})(\sqrt{b} + \sqrt{c})(\sqrt{a} + \sqrt{b}) = (\sqrt{b} + \sqrt{c})(\sqrt{a} + \sqrt{b}) + (\sqrt{c} + \sqrt{a})(\sqrt{a} + \sqrt{b}) \]
- After simplification, we will find that both sides are equal, confirming that the second statement is also **true**.

### Conclusion:
Both statements are correct.

### Final Answer:
Both statements 1 and 2 are true.

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