<p>Let \(n \in N, n > 25\). Let \(A, G, H\) denote the arithmetic mean, geometric mean, and harmonic mean of 25 and \(n\). The least value of \(n\) for which \(A, G, H \in \{25, 26, \ldots, n\}\) is</p>
Step-by-Step Solution
Key Concept: For A, G, H to all be integers in the range [25, n], we need G² = AH (AM-GM-HM relation) and all three means to be integers. Since A = (25+n)/2, G = √(25n), and H = 50n/(25+n), the critical constraint is that √(25n) must be an integer, forcing n = 25k² for some integer k.
<p><strong>Step 1:</strong> Set up the three means.</p><p>A = (25 + n)/2, G = √(25n), H = 50n/(25 + n)</p><p><strong>Step 2:</strong> For G to be an integer, 25n must be a perfect square. Since 25 = 5², we need n = 5²m² = 25m² for some positive integer m. Given n > 25, we need m ≥ 2, so n ∈ {100, 225, 400, ...}</p><p><strong>Step 3:</strong> Check n = 100: A = 125/2 = 62.5 (not an integer). ✗</p><p><strong>Step 4:</strong> For A to be integer, 25 + n must be even. Since 25 is odd, n must be odd. But n = 25m² requires m to be odd (so that 25m² is odd). For m = 2: n = 100 (even). For m = 3: n = 225 (odd). ✓</p><p><strong>Step 5:</strong> Check n = 225: A = (25 + 225)/2 = 125 ✓, G = √(25·225) = 5·15 = 75 ✓, H = 50·225/(25 + 225) = 11250/250 = 45.</p><p><strong>Step 6:</strong> Verify H ∈ [25, 225]: We have 45 ∈ [25, 225]. ✓ All three means are integers in the required range.</p><p>∴ Answer: <strong>D (225)</strong></p>
Correct Answer: D