<p>The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is __________.</p>
Step-by-Step Solution
Key Concept: Let the three consecutive G.P. terms be a/r, a, ar. Use the product condition (a³ = 512) to find a, then apply the A.P. condition after adding 4 to get the common ratio r.
<p><strong>Step 1:</strong> Let the three consecutive G.P. terms be a/r, a, ar where a > 0 and r > 0.</p><p><strong>Step 2:</strong> From the product condition: (a/r) · a · ar = a³ = 512 ⟹ a = 8</p><p><strong>Step 3:</strong> After adding 4 to first and second terms, the three terms are: (8/r + 4), (8 + 4) = 12, and 8r.</p><p><strong>Step 4:</strong> For these to form an A.P., the common difference must be equal:</p><p>12 - (8/r + 4) = 8r - 12</p><p>8 - 8/r = 8r - 12</p><p>20 = 8r + 8/r</p><p>5r = 2r² + 2</p><p>2r² - 5r + 2 = 0</p><p>(2r - 1)(r - 2) = 0</p><p>∴ r = 1/2 or r = 2</p><p><strong>Step 5:</strong> For r = 2: Terms are 4, 8, 16 (verify: 4 + 4 = 8, 8 + 4 = 12, 16 form A.P. with differences 4, 4 ✓)</p><p>For r = 1/2: Terms are 16, 8, 4 (verify: 16 + 4 = 20, 8 + 4 = 12, 4 form A.P. with differences -8, -8 ✓)</p><p><strong>Step 6:</strong> Both solutions are valid. Sum = 4 + 8 + 16 = 28 or 16 + 8 + 4 = 28</p><p>∴ <strong>Answer: 28</strong></p>
Correct Answer: 28