If a, b, c, d, e, f are in AP, then `(e-c)` is equal to which one of the following?

Options

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

Answer

To solve the problem, we need to determine the value of \( e - c \) given that \( a, b, c, d, e, f \) are in an arithmetic progression (AP).

### Step-by-Step Solution:

1. **Understanding the terms in AP**:
In an arithmetic progression, the difference between consecutive terms is constant. Let's denote the first term as \( a \) and the common difference as \( d \). The terms can be expressed as follows:
- \( a = a \)
- \( b = a + d \)
- \( c = a + 2d \)
- \( d = a + 3d \)
- \( e = a + 4d \)
- \( f = a + 5d \)

2. **Identifying the terms**:
We need to find \( e - c \):
\[ e = a + 4d \]
\[ c = a + 2d \]

3. **Calculating \( e - c \)**:
Substitute the expressions for \( e \) and \( c \):
\[ e - c = (a + 4d) - (a + 2d) \]
Simplifying this:
\[ e - c = a + 4d - a - 2d = 4d - 2d = 2d \]

4. **Relating \( d \) to other terms**:
We know that \( d \) can be expressed in terms of other terms in the AP. Specifically, we can express \( d \) as:
\[ d = c - b \]
or
\[ d = e - c \]
Thus, we can write:
\[ e - c = 2d \]

5. **Final Expression**:
Since \( d = c - b \), we can also express \( e - c \) as:
\[ e - c = 2(c - b) \]
Therefore, the final conclusion is:
\[ e - c = 2(d - c) \]

### Conclusion:
Thus, \( e - c \) is equal to \( 2(d - c) \).