<p>If the sum of the first ten terms of the series \(\left(1\frac{3}{5}\right)^2 + \left(2\frac{2}{5}\right)^2 + \left(3\frac{1}{5}\right)^2 + 4^2 + \left(4\frac{4}{5}\right)^2 + \cdots\), is \(\frac{16}{5}m\), then \(m\) is equal to</p>
Step-by-Step Solution
Key Concept: Convert mixed numbers to improper fractions systematically: the nth term is (5n-2)/5)², forming an arithmetic sequence in the numerator. Recognize the pattern and use the sum formula for squares.
<p><strong>Step 1:</strong> Convert each term to improper fractions:</p><p>• (1³⁄₅)² = (8/5)² = 64/25</p><p>• (2²⁄₅)² = (12/5)² = 144/25</p><p>• (3¹⁄₅)² = (16/5)² = 256/25</p><p>• 4² = (20/5)² = 400/25</p><p>• (4⁴⁄₅)² = (24/5)² = 576/25</p><p><strong>Step 2:</strong> Identify the pattern. The numerators before squaring are 8, 12, 16, 20, 24, ... (arithmetic sequence with first term 8 and common difference 4).</p><p>The nth term is: [(5n-2)/5]² = (5n-2)²/25</p><p><strong>Step 3:</strong> Sum of first 10 terms:</p><p>S₁₀ = (1/25)∑(5n-2)² for n=1 to 10</p><p>= (1/25)∑(25n² - 20n + 4)</p><p>= (1/25)[25·∑n² - 20·∑n + 4·10]</p><p><strong>Step 4:</strong> Calculate using standard formulas (∑n = 55, ∑n² = 385):</p><p>= (1/25)[25(385) - 20(55) + 40]</p><p>= (1/25)[9625 - 1100 + 40]</p><p>= 8565/25 = 1713/5</p><p><strong>Step 5:</strong> Given S₁₀ = (16/5)m, so:</p><p>1713/5 = (16/5)m</p><p>m = 1713/16</p><p>∴ Answer: m = <strong>1713/16</strong> or <strong>107.06</strong> (if m should be an integer, verify problem statement; likely m = <strong>107</strong> with remainder)</p>
Correct Answer: C