If a,b,c be in arithmetic progession, then the value of (a+2b-c) (2b+c-a) (a+2b+c), is

Options

A. 16 abc
B. 4 abc
C. 8 abc
D. 3 abc

Answer

To solve the problem, we need to find the value of the expression \( (a + 2b - c)(2b + c - a)(a + 2b + c) \) given that \( a, b, c \) are in arithmetic progression (AP).

### Step-by-Step Solution:

1. **Understanding Arithmetic Progression**:
Since \( a, b, c \) are in AP, we can express the relationship between them. The common difference \( d \) can be defined as:
\[ b - a = c - b \implies 2b = a + c \]

2. **Substituting \( 2b \)**:
From the above relationship, we can substitute \( 2b \) in our expression:
\[ 2b = a + c \]

3. **Simplifying Each Term**:
Now, we will simplify each term in the expression \( (a + 2b - c)(2b + c - a)(a + 2b + c) \):

- **First Term**:
\[ a + 2b - c = a + (a + c) - c = 2a \]

- **Second Term**:
\[ 2b + c - a = (a + c) + c - a = 2c \]

- **Third Term**:
\[ a + 2b + c = a + (a + c) + c = 2a + 2c = 2(a + c) \]

4. **Combining the Terms**:
Now we can substitute these simplified terms back into the expression:
\[ (2a)(2c)(2(a + c)) \]

5. **Calculating the Final Value**:
We can factor out the constants:
\[ = 2 \times 2 \times 2 \times a \times c \times (a + c) = 8a c (a + c) \]

6. **Substituting \( a + c \)**:
We know \( a + c = 2b \), therefore:
\[ 8a c (2b) = 16abc \]

### Final Answer:
Thus, the value of the expression \( (a + 2b - c)(2b + c - a)(a + 2b + c) \) is:
\[ \boxed{16abc} \]