Question:

Step 1: Set up the arithmetic progression with known terms. Let the A.P. have first term $l$, common difference $d$, and $n$ total terms. We are given: - First term: $a_1 = l$ - Fifth term: $a_5 = m$ - Last term: $a_n = p$ Step 2: Find the common difference using the fifth term. The fifth term of an A.P. is given by: $$a_5 = l + 4d = m$$ Solving for $d$: $$d = \frac{m-l}{4}$$ Step 3: Find the number of terms using the last term. The last term of an A.P. is given by: $$a_n = l + (n-1)d = p$$ Rearranging: $$(n-1)d = p - l$$ $$n - 1 = \frac{p-l}{d}$$ Substituting $d = \frac{m-l}{4}$: $$n - 1 = \frac{p-l}{\frac{m-l}{4}} = \frac{4(p-l)}{m-l}$$ Step 4: Simplify to find $n$. $$n = 1 + \frac{4(p-l)}{m-l}$$ $$n = \frac{m-l + 4(p-l)}{m-l}$$ $$n = \frac{m - l + 4p - 4l}{m-l}$$ $$n = \frac{4p + m - 5l}{m-l}$$ Step 5: Calculate the sum of the A.P. The sum of an A.P. is: $$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(l + p)$$ Substituting the value of $n$: $$S_n = \frac{1}{2} \cdot \frac{4p + m - 5l}{m-l} \cdot (l+p)$$ $$S_n = \frac{(l+p)(4p+m-5l)}{2(m-l)}$$ Step 6: Compare with the given expression to find $k$. We are given that the sum equals: $$S_n = \frac{(l+p)(4p+m-5l)}{k(m-l)}$$ Comparing our result with the given expression: $$\frac{(l+p)(4p+m-5l)}{2(m-l)} = \frac{(l+p)(4p+m-5l)}{k(m-l)}$$ Therefore: $$k = 2$$ The answer is **Option 1: 2**