If the first, fifth and last terms of an `A.P.` is `l`, `m`, `p`, respectively, and sum of the `A.P.` is `((l+p)(4p+m-5l))/(k(m-l))` then `k` is

Options

A. `2`
B. `3`
C. `4`
D. `5`

Explanation

To solve the problem step by step, we will use the properties of Arithmetic Progression (A.P.).

### Step 1: Identify the terms of the A.P.
Given:
- First term \( a = l \)
- Fifth term \( a + 4d = m \)
- Last term \( a_n = p \)

### Step 2: Express the fifth term in terms of \( d \)
From the fifth term, we can express \( d \):
\[ l + 4d = m \]
Rearranging gives:
\[ 4d = m - l \quad \Rightarrow \quad d = \frac{m - l}{4} \]

### Step 3: Express the last term in terms of \( n \) and \( d \)
The last term can be expressed as:
\[ p = a + (n-1)d \]
Substituting \( a \) and \( d \):
\[ p = l + (n-1)\left(\frac{m - l}{4}\right) \]

### Step 4: Rearrange to find \( n \)
Rearranging the equation gives:
\[ p = l + \frac{(n-1)(m - l)}{4} \]
Multiplying through by 4 to eliminate the fraction:
\[ 4p = 4l + (n-1)(m - l) \]
This simplifies to:
\[ 4p - 4l = (n-1)(m - l) \]
Now, solving for \( n \):
\[ n - 1 = \frac{4p - 4l}{m - l} \]
Thus,
\[ n = \frac{4p - 4l}{m - l} + 1 \]

### Step 5: Simplify \( n \)
Combining the terms:
\[ n = \frac{4p - 4l + m - l}{m - l} = \frac{4p + m - 5l}{m - l} \]

### Step 6: Find the sum of the A.P.
The sum \( S_n \) of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2}(a + a_n) = \frac{n}{2}(l + p) \]
Substituting \( n \):
\[ S_n = \frac{\frac{4p + m - 5l}{m - l}}{2}(l + p) = \frac{(4p + m - 5l)(l + p)}{2(m - l)} \]

### Step 7: Set the two expressions for the sum equal
According to the problem, the sum is also given by:
\[ S_n = \frac{(l + p)(4p + m - 5l)}{k(m - l)} \]
Setting the two expressions for \( S_n \) equal:
\[ \frac{(4p + m - 5l)(l + p)}{2(m - l)} = \frac{(l + p)(4p + m - 5l)}{k(m - l)} \]

### Step 8: Cancel common terms
Cancelling \( (l + p)(4p + m - 5l) \) from both sides (assuming \( l + p \neq 0 \)):
\[ \frac{1}{2(m - l)} = \frac{1}{k(m - l)} \]

### Step 9: Solve for \( k \)
Cross-multiplying gives:
\[ k(m - l) = 2(m - l) \]
Assuming \( m - l \neq 0 \), we can divide both sides by \( m - l \):
\[ k = 2 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{2} \]