<p>If the sum of the first ten terms of the series \(\left(1\dfrac{3}{5}\right)^2 + \left(2\dfrac{2}{5}\right)^2 + \left(3\dfrac{1}{5}\right)^2 + 4^2 + \left(4\dfrac{4}{5}\right)^2 + \ldots\) is \(\dfrac{16}{5}m\), then \(m\) is equal to</p>
Step-by-Step Solution
Key Concept: Convert mixed numbers to improper fractions: 1³⁄₅ = 8/5, 2²⁄₅ = 12/5, 3¹⁄₅ = 16/5, etc., revealing that the nth term is ((4n+4)/5)² = (4(n+1)/5)². Use the formula Σk² = n(n+1)(2n+1)/6 to find the sum efficiently.
<p><strong>Step 1: Convert to improper fractions</strong><br>1³⁄₅ = 8/5, 2²⁄₅ = 12/5, 3¹⁄₅ = 16/5, 4 = 20/5, 4⁴⁄₅ = 24/5, ...</p><p><strong>Step 2: Identify the pattern</strong><br>The nth term: aₙ = (4(n+1)/5)² = 16(n+1)²/25</p><p><strong>Step 3: Sum the first 10 terms</strong><br>S₁₀ = Σₙ₌₁¹⁰ 16(n+1)²/25 = (16/25)Σₙ₌₁¹⁰(n+1)²</p><p><strong>Step 4: Reindex and compute</strong><br>Σₙ₌₁¹⁰(n+1)² = Σₖ₌₂¹¹k² = (Σₖ₌₁¹¹k²) - 1 = [11(12)(23)/6] - 1 = 506 - 1 = 505</p><p><strong>Step 5: Calculate final sum</strong><br>S₁₀ = (16/25) × 505 = 8080/25 = 1616/5</p><p><strong>Step 6: Match with given form</strong><br>Given: S₁₀ = (16/5)m<br>1616/5 = (16/5)m<br>m = 1616/16 = 101</p><p>∴ Answer: <strong>m = 101</strong></p>
Correct Answer: D