<p><strong>162.</strong> If the first, fifth and last terms of an A.P. are \(l, m, p\) respectively and the sum of A.P. is \(\dfrac{(l+p)(4p+m-5l)}{k(m-l)}\), then \(k\) is:</p>
Step-by-Step Solution
Key Concept: Express the number of terms n and common difference d in terms of l, m, p using the given first, fifth, and last terms. Then substitute into the sum formula S = n/2(first + last) and match coefficients to find k.
<p><strong>Step 1:</strong> From the A.P., let first term = l, common difference = d, number of terms = n.</p><p>Given: a₁ = l, a₅ = m, aₙ = p</p><p><strong>Step 2:</strong> From a₅: l + 4d = m, so d = (m - l)/4</p><p><strong>Step 3:</strong> From aₙ: l + (n-1)d = p</p><p>Substituting d: l + (n-1)·(m-l)/4 = p</p><p>(n-1)·(m-l)/4 = p - l</p><p>n - 1 = 4(p-l)/(m-l)</p><p>n = 1 + 4(p-l)/(m-l) = [m - l + 4p - 4l]/(m-l) = (4p + m - 5l)/(m-l)</p><p><strong>Step 4:</strong> Sum of A.P.: S = n/2(a₁ + aₙ) = n/2(l + p)</p><p>S = [(4p + m - 5l)/(m-l)] · (1/2)(l + p)</p><p>S = (l + p)(4p + m - 5l) / [2(m-l)]</p><p><strong>Step 5:</strong> Comparing with given sum = (l+p)(4p+m-5l) / [k(m-l)]:</p><p>k(m-l) = 2(m-l)</p><p>∴ k = <strong>2</strong></p>
Correct Answer: A