If a, b, c, d, e are in A.P. then the value of a - 4b + 6c - 4d + e is

Options

A. 0
B. 1
C. 2
D. none

Answer

To solve the problem, we need to find the value of \( a - 4b + 6c - 4d + e \) given that \( a, b, c, d, e \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding A.P.**:
Since \( a, b, c, d, e \) are in A.P., we can express them in terms of the first term \( a \) and the common difference \( d \):
- \( a = a \)
- \( b = a + d \)
- \( c = a + 2d \)
- \( d = a + 3d \)
- \( e = a + 4d \)

2. **Substituting Values**:
We substitute the values of \( b, c, d, e \) into the expression \( a - 4b + 6c - 4d + e \):
\[ a - 4(a + d) + 6(a + 2d) - 4(a + 3d) + (a + 4d) \]

3. **Expanding the Expression**:
Now, we expand the expression:
\[ = a - 4a - 4d + 6a + 12d - 4a - 12d + a + 4d \]

4. **Combining Like Terms**:
Combine all the terms involving \( a \) and \( d \):
- For \( a \):
\[ a - 4a + 6a - 4a + a = 0 \]
- For \( d \):
\[ -4d + 12d - 12d + 4d = 0 \]

5. **Final Result**:
Therefore, the entire expression simplifies to:
\[ 0 + 0 = 0 \]

Thus, the value of \( a - 4b + 6c - 4d + e \) is **0**.