If a, b, c, d are distinct positive numbers in A.P., then:

Options

A. a+d = b+c
B. a + c = b + d
C. a + d = 2 (b + c)
D. a + d = 3 (b + c)

Answer

To solve the problem, we need to establish the relationships among the distinct positive numbers \( a, b, c, d \) that are in Arithmetic Progression (A.P.). Let's denote the first term as \( a \) and the common difference as \( D \). The terms can be expressed as follows:

1. \( a = a \) (first term)
2. \( b = a + D \) (second term)
3. \( c = a + 2D \) (third term)
4. \( d = a + 3D \) (fourth term)

Now, we will analyze the given options to determine which condition is true.

### Step 1: Analyze the first option \( a + D = b + c \)

**Left Hand Side:**
\[ a + D = a + D \]

**Right Hand Side:**
\[ b + c = (a + D) + (a + 2D) = 2a + 3D \]

Now, we compare both sides:
\[ a + D \neq 2a + 3D \]
This means the first option is not true.

### Step 2: Analyze the second option \( a + c = 2b \)

**Left Hand Side:**
\[ a + c = a + (a + 2D) = 2a + 2D \]

**Right Hand Side:**
\[ 2b = 2(a + D) = 2a + 2D \]

Now, we compare both sides:
\[ a + c = 2b \]
This means the second option is true.

### Step 3: Analyze the third option \( a + d = 2c \)

**Left Hand Side:**
\[ a + d = a + (a + 3D) = 2a + 3D \]

**Right Hand Side:**
\[ 2c = 2(a + 2D) = 2a + 4D \]

Now, we compare both sides:
\[ a + d \neq 2c \]
This means the third option is not true.

### Step 4: Analyze the fourth option \( b + d = 2c \)

**Left Hand Side:**
\[ b + d = (a + D) + (a + 3D) = 2a + 4D \]

**Right Hand Side:**
\[ 2c = 2(a + 2D) = 2a + 4D \]

Now, we compare both sides:
\[ b + d = 2c \]
This means the fourth option is true.

### Conclusion:
The true conditions among the options are:
- \( a + c = 2b \)
- \( b + d = 2c \)