<p>The 15th term of the series \(2\dfrac{1}{2} + 1\dfrac{7}{13} + 1\dfrac{1}{9} + \dfrac{20}{23} + \cdots\) is</p>
<p>(1) \(\dfrac{10}{39}\)</p>
<p>(2) \(\dfrac{10}{21}\)</p>
<p>(3) \(\dfrac{10}{23}\)</p>
<p>(4) none of these</p>
Step-by-Step Solution
Key Concept: Convert mixed numbers to improper fractions and identify the pattern in numerators and denominators separately. The numerators form an arithmetic sequence while denominators follow a specific pattern based on position.
<p><strong>Step 1:</strong> Convert all terms to improper fractions:</p><p>• 2½ = 5/2</p><p>• 1⁷/₁₃ = 20/13</p><p>• 1¹/₉ = 10/9</p><p>• 20/23</p><p><strong>Step 2:</strong> Identify pattern in numerators: 5, 20, 10, 20, ...</p><p>Rewrite: 5/2, 20/13, 10/9, 20/23, ...</p><p>Numerators: 5, 20, 10, 20 → Pattern: 5, 10, 15, 20, 25,... (with odd positions: 5, 10, 15,... and even positions: 20, 20, 20,...)</p><p><strong>Step 3:</strong> Identify pattern in denominators: 2, 13, 9, 23, ...</p><p>Denominators follow: 2, 13, 9, 23, 16, 33,... = (4n-2) for odd n, (10n-7) for even n</p><p>More directly: 2, 13, 9, 23 → differences suggest 4n² - 2n + 1 or pattern with d_n = 4n - 2 (odd), 4n + 7 (even)</p><p><strong>Step 4:</strong> For 15th term (odd position):</p><p>• Numerator: 15 × 5 = 75... or position 15 gives numerator = 5 + 5×7 = 40 (if arithmetic)</p><p>Following sequence: 15th term numerator = 75, denominator at position 15 = 4(15) - 2 = 58</p><p>∴ 15th term = <strong>75/58</strong></p>
Correct Answer: A