If a,b,c,d `in R^(+)` such that a,b,c and d are in H.P. then

Options

A. `a+dgtb+c`
B. `a+bgtc+d`
C. `a+cgtb+d`
D. none of these

Answer

To solve the problem where \( a, b, c, d \) are in Harmonic Progression (H.P.), we start by recalling the definition of H.P. and the properties associated with it.

### Step-by-Step Solution:

1. **Understanding Harmonic Progression**:
If \( a, b, c, d \) are in H.P., then the 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 middle terms must satisfy the condition:
\[ 2b = a + c \quad \text{(1)} \]
and
\[ 2c = b + d \quad \text{(2)} \]

3. **Using Equation (1)**:
From equation (1), we can rearrange it to express \( a \) in terms of \( b \) and \( c \):
\[ a = 2b - c \quad \text{(3)} \]

4. **Using Equation (2)**:
Similarly, from equation (2), we can express \( d \) in terms of \( b \) and \( c \):
\[ d = 2c - b \quad \text{(4)} \]

5. **Adding Equations (3) and (4)**:
Now, we add equations (3) and (4):
\[ a + d = (2b - c) + (2c - b) \]
Simplifying this gives:
\[ a + d = 2b - c + 2c - b = b + c \]

6. **Final Inequality**:
Therefore, we conclude that:
\[ a + d = b + c \]
This means that \( a + d \) is equal to \( b + c \).

### Conclusion:
Since the problem states that \( a, b, c, d \) are in H.P., we have derived that:
\[ a + d = b + c \]
Thus, the correct relationship is \( a + d = b + c \).