The first and last term of an A.P. are a and l respectively. If S be the sum of all the terms of the A.P., them the common difference is

Options

A. `(l^(2)-a^(2))/(2S-(l+a))`
B. `(l^(2)-a^(2))/(2S-(l-a))`
C. `(l^(2)+a^(2))/(2S+(l+a))`
D. `(l^(2)+a^(2))/(2S-(l+a))`

Explanation

To find the common difference \( D \) of an arithmetic progression (A.P.) where the first term is \( a \), the last term is \( l \), and the sum of all terms is \( S \), we can follow these steps:

### Step 1: Identify the number of terms \( N \)
In an A.P., the last term \( l \) can be expressed in terms of the first term \( a \), the common difference \( D \), and the number of terms \( N \):
\[ l = a + (N - 1)D \]
From this equation, we can rearrange it to express \( D \):
\[ (N - 1)D = l - a \quad \Rightarrow \quad D = \frac{l - a}{N - 1} \]

### Step 2: Use the formula for the sum of an A.P.
The sum \( S \) of the first \( N \) terms of an A.P. is given by the formula:
\[ S = \frac{N}{2} (a + l) \]
From this, we can express \( N \):
\[ N = \frac{2S}{a + l} \]

### Step 3: Substitute \( N \) into the equation for \( D \)
Now we can substitute the expression for \( N \) back into the equation for \( D \):
\[ D = \frac{l - a}{\frac{2S}{a + l} - 1} \]
To simplify this, we find a common denominator:
\[ D = \frac{l - a}{\frac{2S - (a + l)}{a + l}} = \frac{(l - a)(a + l)}{2S - (a + l)} \]

### Step 4: Final expression for the common difference \( D \)
Thus, the common difference \( D \) can be expressed as:
\[ D = \frac{(l - a)(a + l)}{2S - (a + l)} \]

### Summary
The common difference \( D \) of the A.P. is given by:
\[ D = \frac{(l - a)(a + l)}{2S - (a + l)} \]