<p>Find four numbers in G.P. whose sum is 85 and product is 4096.</p>
Step-by-Step Solution
Key Concept: Use the property that for four terms in G.P., if we denote them as a/r³, a/r, ar, ar³, their product gives a⁴ = 4096 directly, and symmetry simplifies the sum equation to a(1/r³ + 1/r + r + r³) = 85.
<p><strong>Step 1:</strong> Let the four numbers in G.P. be a/r³, a/r, ar, ar³ (symmetric form centered at a).</p><p><strong>Step 2:</strong> Product condition: (a/r³)(a/r)(ar)(ar³) = a⁴ = 4096 → a = ±8</p><p><strong>Step 3:</strong> Sum condition: a/r³ + a/r + ar + ar³ = 85. Factor as a(1/r³ + 1/r + r + r³) = 85</p><p><strong>Step 4:</strong> Let t = r + 1/r. Then 1/r³ + r³ = t³ - 3t, so: a(t³ - 3t + t) = a(t³ - 2t) = 85</p><p><strong>Step 5:</strong> For a = 8: 8t(t² - 2) = 85. Testing t = 5/2: 8(5/2)(25/4 - 2) = 8(5/2)(17/4) = 85 ✓</p><p><strong>Step 6:</strong> From r + 1/r = 5/2: 2r² - 5r + 2 = 0 → r = 2 or r = 1/2</p><p><strong>Step 7:</strong> For r = 2: a/r³ = 1, a/r = 2, ar = 16, ar³ = 64</p><p><strong>Step 8:</strong> For r = 1/2: a/r³ = 64, a/r = 16, ar = 4, ar³ = 1</p><p><strong>Step 9:</strong> For a = -8: similar analysis gives negative progressions (typically not the primary answer for such problems).</p><p>∴ Answer: <strong>1, 4, 16, 64</strong> or <strong>64, 16, 4, 1</strong></p>
Correct Answer: 1, 4, 16, 64 or 64, 16, 4, 1