If four positive numbers a, b, c, d are in H.P. then which one of the following is true.

Options

A. `a^(n) + c^(n) gt 2b^(n)`
B. `a + d gt b + c`
C. `(a//b) gt (c//d)`
D. `b^(n) + d^(n) gt 2c^(n)`

Answer

To determine which statements are true if four positive numbers \( a, b, c, d \) are in Harmonic Progression (H.P.), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Harmonic Progression**:
If four numbers \( a, b, c, d \) are in H.P., then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d} \) are in Arithmetic Progression (A.P.).

2. **Setting Up the A.P. Condition**:
For numbers to be in A.P., the condition is:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]
and
\[ 2 \cdot \frac{1}{c} = \frac{1}{b} + \frac{1}{d} \]

3. **Cross-Multiplying the A.P. Conditions**:
From the first condition:
\[ 2bc = ac + ab \quad \text{(1)} \]
From the second condition:
\[ 2bc = bd + cd \quad \text{(2)} \]

4. **Rearranging the Equations**:
From equation (1):
\[ ab + ac < 2bc \quad \text{(since b and c are positive)} \]
Rearranging gives:
\[ a + d > b + c \]
This implies:
\[ a + d > b + c \quad \text{(Option 2)} \]

5. **Using the Product of Terms**:
From the rearranged inequalities, we can also analyze:
\[ ad > bc \quad \text{(Option 3)} \]
This can be derived from the properties of the A.P. and the positivity of the numbers.

6. **Conclusion**:
Both \( a + d > b + c \) and \( ad > bc \) are true statements when \( a, b, c, d \) are in H.P.

### Final Answer:
- The true statements are:
- \( a + d > b + c \) (Option 2)
- \( ad > bc \) (Option 3)