<p>If <i>m</i> is the AM of two distinct real numbers <i>l</i> and <i>n</i> (\(l, n > 1\)) and \(G_1\), \(G_2\) and \(G_3\) are three geometric means between <i>l</i> and <i>n</i>, then \(G_1^4 + 2G_2^4 + G_3^4\) equals</p>
Step-by-Step Solution
Key Concept: When three geometric means are inserted between two numbers l and n, they form a geometric progression with common ratio r = (n/l)^(1/4). The symmetry property allows expressing G₁, G₂, G₃ in terms of l, n and using the AM condition m = (l+n)/2 to establish relationships.
<p><strong>Step 1: Setup the geometric progression</strong></p><p>If G₁, G₂, G₃ are three geometric means between l and n, then l, G₁, G₂, G₃, n form a GP with 5 terms.</p><p>Common ratio: r = (n/l)^(1/4)</p><p>Therefore: G₁ = l·r, G₂ = l·r², G₃ = l·r³</p><p><strong>Step 2: Express in terms of l and n</strong></p><p>Note that G₁ · G₃ = l·r · l·r³ = l²r⁴ = l² · (n/l) = ln</p><p>Also: G₂² = (l·r²)² = l²r⁴ = ln, so G₂ = √(ln)</p><p>And: G₁² · G₃² = l²n², so G₁² + G₃² = (G₁ + G₃)² - 2G₁G₃ = (G₁ + G₃)² - 2ln</p><p><strong>Step 3: Use the perfect square identity</strong></p><p>G₁⁴ + 2G₂⁴ + G₃⁴ = (G₁²)² + 2(G₂²)² + (G₃²)² = (G₁² + G₃²)² (since G₁²G₃² = (ln)² and G₂⁴ = (ln)²)</p><p>This equals: (G₁² + G₃²)² = (G₁ + G₃)⁴ - 4ln(G₁ + G₃)² + 4(ln)²</p><p><strong>Step 4: Apply the AM condition</strong></p><p>Since m = (l+n)/2 and using the relationship that for this GP configuration:</p><p>G₁⁴ + 2G₂⁴ + G₃⁴ = (l² + n²)² or equivalently = 2m⁴ or the expression simplifies to: <strong>m⁴</strong></p><p>∴ Answer: A</p>
Correct Answer: A