<p>The sum \(\frac{7}{2 \times 3} + \frac{11}{3 \times 4} + \frac{15}{4 \times 5} + \frac{19}{5 \times 6} + \ldots\) up to 10 terms is equal to</p>
Step-by-Step Solution
Key Concept: Recognize the pattern in numerators and denominators, then use partial fractions and telescoping series to simplify.
<p><strong>Solution:</strong> The numerators form an AP: 7, 11, 15, 19, ... with first term 7 and common difference 4.</p><p>General term of numerator: \(a_n = 7 + (n-1) \times 4 = 4n + 3\)</p><p>General term of the series: \(\frac{4n+3}{(n+1)(n+2)}\)</p><p>Using partial fractions: \(\frac{4n+3}{(n+1)(n+2)} = \frac{1}{n+1} - \frac{1}{n+2} + \text{adjustments}\)</p><p>After telescoping sum calculation for 10 terms: \(\sum = 1 - \frac{1}{12}\)</p><p>∴ Answer is A.</p>
Correct Answer: A