<p>If \(m\) is the A.M. of two distinct real numbers \(l\) and \(n\) \((l, n > 1)\) and \(G_1, G_2\) and \(G_3\) are three geometric means between \(l\) and \(n\), then \((G_1)^4 + 2(G_2)^4 + (G_3)^4\) equals</p>
Step-by-Step Solution
Key Concept: When three geometric means are inserted between l and n, they form a G.P. with common ratio r = (n/l)^(1/4). The key is expressing G₁, G₂, G₃ in terms of l, r and using the A.M. condition m = (l+n)/2 to create a relationship that simplifies the fourth power expression.
<p><strong>Step 1:</strong> Set up the G.P. between l and n with three geometric means inserted.</p><p>The sequence is: l, G₁, G₂, G₃, n (5 terms in G.P.)</p><p>Common ratio: r = (n/l)^(1/4)</p><p>Therefore: G₁ = lr^1, G₂ = lr^2, G₃ = lr^3, where r⁴ = n/l</p><p><strong>Step 2:</strong> Express in terms of roots.</p><p>G₁ = l·(n/l)^(1/4) = l^(3/4)·n^(1/4)</p><p>G₂ = l^(1/2)·n^(1/2) = √(ln)</p><p>G₃ = l^(1/4)·n^(3/4)</p><p><strong>Step 3:</strong> Calculate fourth powers.</p><p>(G₁)⁴ = l³n</p><p>(G₂)⁴ = (ln)²</p><p>(G₃)⁴ = ln³</p><p><strong>Step 4:</strong> Substitute into the expression.</p><p>(G₁)⁴ + 2(G₂)⁴ + (G₃)⁴ = l³n + 2(ln)² + ln³</p><p>= l³n + 2l²n² + ln³</p><p>= ln(l² + 2ln + n²)</p><p>= ln(l + n)²</p><p><strong>Step 5:</strong> Use the A.M. condition.</p><p>Given: m = (l+n)/2, so l+n = 2m</p><p>(G₁)⁴ + 2(G₂)⁴ + (G₃)⁴ = ln(2m)² = 4m²ln</p><p>∴ Answer: <strong>4m²ln</strong> (or equivalent form matching option B)</p>
Correct Answer: B