<p>Let <em>n</em> ∈ ℕ, <em>n</em> ≠ 25. If <em>A</em>, <em>G</em> and <em>H</em> denote the arithmetic mean, geometric mean and harmonic mean of 25 and <em>n</em>. Then, the least value of <em>n</em> for which <em>A</em>, <em>G</em>, <em>H</em> ∈ {25, 26, ..., <em>n</em>}, is</p>
Step-by-Step Solution
Key Concept: We need all three means (A, G, H) to lie in the interval [25, n]. Using the AM-GM-HM inequality and algebraic constraints, we find the minimum n satisfying all conditions.
<p><strong>Step 1: Express the means.</strong> For 25 and n where n ≠ 25:</p><p>A = (25 + n)/2</p><p>G = √(25n)</p><p>H = 2(25n)/(25 + n)</p><p><strong>Step 2: Apply the AM-GM-HM inequality.</strong> We know H ≤ G ≤ A (with equality iff 25 = n).</p><p><strong>Step 3: Constraint that A, G, H ∈ {25, 26, ..., n}.</strong> Since A > G > H for n ≠ 25, we need:</p><p>• H ≥ 25 (H must be at least 25)</p><p>• A ≤ n (A must not exceed n)</p><p><strong>Step 4: Check constraint A ≤ n.</strong> We have (25 + n)/2 ≤ n, which gives 25 + n ≤ 2n, so n ≥ 25. This is always satisfied for n > 25.</p><p><strong>Step 5: Check the binding constraint H ≥ 25.</strong> We need:</p><p>2(25n)/(25 + n) ≥ 25</p><p>50n ≥ 25(25 + n)</p><p>50n ≥ 625 + 25n</p><p>25n ≥ 625</p><p>n ≥ 25</p><p>This is satisfied, but we need H to be an integer in {25, 26, ..., n}.</p><p><strong>Step 6: Check that G is an integer.</strong> For G = √(25n) to be an integer, 25n must be a perfect square. Since 25 = 5², we need n = k² for some integer k ≥ 5 (so n > 25).</p><p><strong>Step 7: Check that H is an integer.</strong> For n = k², we have H = 50k²/(25 + k²). For H to be an integer, (25 + k²) must divide 50k².</p><p><strong>Step 8: Test the options.</strong></p><p>• n = 49: k = 7, H = 50(49)/(25 + 49) = 2450/74 = 1225/37 (not integer)</p><p>• n = 81: k = 9, H = 50(81)/(25 + 81) = 4050/106 = 2025/53 (not integer)</p><p>• n = 169: k = 13, H = 50(169)/(25 + 169) = 8450/194 = 4225/97 (not integer)</p><p>• n = 225: k = 15, H = 50(225)/(25 + 225) = 11250/250 = 45 ✓</p><p>Check: A = (25 + 225)/2 = 125, G = √(25×225) = √5625 = 75, H = 45</p><p>All three values {45, 75, 125} ⊂ {25, 26, ..., 225} ✓</p><p><strong>∴ Answer: d</strong></p>
Correct Answer: d