<p>The sum of the 3rd and 4th terms of a G.P. is 60 and the product of its first three terms is 1000. If the first term of this G.P. is positive, then its 7th term is</p>
Step-by-Step Solution
Key Concept: Use the property that in a G.P., if the first term is 'a' and common ratio is 'r', then the middle term of any three consecutive terms is the geometric mean of the outer terms. This helps us connect the product condition to individual terms elegantly.
<p><strong>Step 1:</strong> Let the G.P. have first term <em>a</em> and common ratio <em>r</em>. The terms are: <em>a, ar, ar², ar³, ...</em></p><p><strong>Step 2:</strong> Product of first three terms: <em>a · ar · ar²</em> = <em>a³r³</em> = 1000<br>Therefore: <em>(ar)³</em> = 1000 ⟹ <em>ar</em> = 10 (taking real cube root)</p><p><strong>Step 3:</strong> Sum of 3rd and 4th terms: <em>ar²</em> + <em>ar³</em> = 60<br>Factor: <em>ar²(1 + r)</em> = 60<br>Substitute <em>ar</em> = 10: <em>10r(1 + r)</em> = 60<br>⟹ <em>r(1 + r)</em> = 6<br>⟹ <em>r² + r - 6</em> = 0<br>⟹ <em>(r + 3)(r - 2)</em> = 0<br>So <em>r</em> = 2 or <em>r</em> = -3</p><p><strong>Step 4:</strong> From <em>ar</em> = 10:<br>If <em>r</em> = 2: <em>a</em> = 5 (positive ✓)<br>If <em>r</em> = -3: <em>a</em> = -10/3 (negative ✗)</p><p><strong>Step 5:</strong> With <em>a</em> = 5 and <em>r</em> = 2:<br>7th term = <em>ar⁶</em> = 5 × 2⁶ = 5 × 64 = <strong>320</strong></p><p><strong>Verification:</strong> 3rd term = 20, 4th term = 40, sum = 60 ✓<br>Product of first three: 5 × 10 × 20 = 1000 ✓</p><p>∴ Answer: <strong>320</strong></p>
Correct Answer: A