The first term, second term and middle term of an A.P. are a, b and c respectively. The sum of this A.P. is :

Options

A. `(2b(c-a))/(b-a)`
B. `(2a(a-b))/(b-c)`
C. `(2c(c-a))/(b-a)`
D. None of these

Explanation

To find the sum of the arithmetic progression (A.P.) given that the first term is \( a \), the second term is \( b \), and the middle term is \( c \), we can follow these steps:

### Step 1: Understand the terms of the A.P.
In an A.P., the first term is \( a \), the second term is \( b \), and the middle term is \( c \). Since \( b \) is the second term, we can express it in terms of \( a \) and the common difference \( d \):
\[ b = a + d \]

### Step 2: Express the middle term.
The middle term \( c \) can be expressed as:
\[ c = \frac{a + b}{2} \]
Substituting \( b \) from the previous step:
\[ c = \frac{a + (a + d)}{2} = \frac{2a + d}{2} \]

### Step 3: Solve for the common difference \( d \).
From the equation for \( c \), we can rearrange it to find \( d \):
\[ 2c = 2a + d \implies d = 2c - 2a \]

### Step 4: Determine the number of terms in the A.P.
Since we have the first term, second term, and a middle term, we can assume there are \( n \) terms in total. Since \( c \) is the middle term, \( n \) must be odd. Let’s denote the total number of terms as \( n \).

### Step 5: Find the sum of the A.P.
The sum \( S_n \) of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (\text{first term} + \text{last term}) \]
The last term can be expressed as:
\[ \text{last term} = a + (n-1)d \]
Substituting \( d \) from Step 3:
\[ \text{last term} = a + (n-1)(2c - 2a) = a + (n-1)(2c) - (n-1)(2a) \]
Simplifying this gives:
\[ \text{last term} = a + 2c(n-1) - 2a(n-1) = (1 - 2(n-1))a + 2c(n-1) \]

### Step 6: Substitute back into the sum formula.
Now substituting the first term and last term into the sum formula:
\[ S_n = \frac{n}{2} \times \left( a + \left( (1 - 2(n-1))a + 2c(n-1) \right) \right) \]
This will give us the final expression for the sum of the A.P.

### Final Result
After simplification, we can express the sum \( S_n \) in terms of \( a \), \( b \), and \( c \).