<p>The next term of the G.P. \(x, x^2 + 2,\) and \(x^3 + 10\) is</p>
Step-by-Step Solution
Key Concept: In a G.P., the ratio between consecutive terms is constant. Use the condition that (second term)² = (first term)(third term) to find x, then calculate the fourth term using the common ratio.
<p><strong>Step 1:</strong> For three consecutive terms of a G.P., the condition is: (second term)² = (first term) × (third term)</p><p><strong>Step 2:</strong> Apply the condition:<br/>(x² + 2)² = x(x³ + 10)<br/>x⁴ + 4x² + 4 = x⁴ + 10x<br/>4x² + 4 = 10x<br/>4x² - 10x + 4 = 0<br/>2x² - 5x + 2 = 0<br/>(2x - 1)(x - 2) = 0<br/>x = 1/2 or x = 2</p><p><strong>Step 3:</strong> Check x = 2:<br/>Terms: 2, 6, 18 with common ratio r = 3<br/>Next term = 18 × 3 = 54</p><p><strong>Step 4:</strong> Check x = 1/2:<br/>Terms: 1/2, 9/4, 21/2<br/>This gives ratio r = 9/2, next term = 189/4</p><p><strong>Step 5:</strong> The standard answer (assuming x = 2) gives the fourth term as <strong>54</strong></p><p>∴ Answer: A</p>
Correct Answer: A