If a,b,c are in AP and `(a+b)/(2) = x , (b+c)/(2) = y` , then the value of `(x+y)` is .

Options

A. 2a
B. 2b
C. `2(a+b)`
D. `2(b+c)`

Answer

To solve the problem, we need to find the value of \( x + y \) given that \( a, b, c \) are in arithmetic progression (AP) and the definitions of \( x \) and \( y \).

### Step-by-Step Solution:

1. **Understanding the Arithmetic Progression (AP)**:
Since \( a, b, c \) are in AP, we know that:
\[ 2b = a + c \]
This is a fundamental property of numbers in AP.

2. **Expressing \( x \) and \( y \)**:
We are given:
\[ x = \frac{a + b}{2} \]
\[ y = \frac{b + c}{2} \]

3. **Finding \( x + y \)**:
We can add \( x \) and \( y \):
\[ x + y = \frac{a + b}{2} + \frac{b + c}{2} \]
Combining the fractions:
\[ x + y = \frac{(a + b) + (b + c)}{2} = \frac{a + 2b + c}{2} \]

4. **Substituting \( a + c \)**:
From the property of AP, we know \( a + c = 2b \). Therefore, we can substitute this into our equation:
\[ x + y = \frac{2b + 2b}{2} = \frac{4b}{2} = 2b \]

5. **Final Answer**:
Thus, the value of \( x + y \) is:
\[ x + y = 2b \]