<p>The value of \ \left(\frac{8}{5}\right)^2 + \left(\frac{12}{5}\right)^2 + \left(\frac{16}{5}\right)^2 + \left(\frac{20}{5}\right)^2 + \cdots\ up to 10 terms is \(\frac{16}{5}m\). Find \(m\).</p>
Step-by-Step Solution
Key Concept: Recognize this as a sum of squares of an arithmetic sequence. Factor out the common denominator squared, identify the numerators as 8, 12, 16, 20,... (arithmetic progression with first term 8 and common difference 4), then use the formula for sum of squares: Σ(a + (n-1)d)² = na² + 2ad·Σ(n-1) + d²·Σ(n-1)².
<p><strong>Step 1:</strong> Identify the pattern in numerators: 8, 12, 16, 20, ... This is an A.P. with first term a = 8 and common difference d = 4. The general term is 8 + (n-1)·4 = 4n + 4 = 4(n+1).</p><p><strong>Step 2:</strong> Rewrite the sum as: <br>∑(k=1 to 10) [4(k+1)/5]² = (16/25)∑(k=1 to 10)(k+1)²</p><p><strong>Step 3:</strong> Compute ∑(k=1 to 10)(k+1)² = ∑(j=2 to 11)j² = [∑(j=1 to 11)j²] - 1<br>Using ∑j² = n(n+1)(2n+1)/6: ∑(j=1 to 11)j² = 11·12·23/6 = 506<br>So ∑(j=2 to 11)j² = 506 - 1 = 505</p><p><strong>Step 4:</strong> The sum equals (16/25)·505 = (16·505)/25 = (16·101)/5 = (16/5)·101</p><p><strong>Step 5:</strong> Comparing with (16/5)m, we get m = 101.</p><p>∴ Answer: <strong>101</strong></p>
Correct Answer: 101