Question:

<p><strong>Step 1:</strong> Factor the denominator using difference of squares.</p><p>r⁴ + 1/4 = (r²)² + (1/2)² = (r² - r + 1/2)(r² + r + 1/2)</p><p><strong>Step 2:</strong> Set up partial fractions.</p><p>$$\frac{r}{(r² - r + 1/2)(r² + r + 1/2)} = \frac{A}{r² - r + 1/2} + \frac{B}{r² + r + 1/2}$$</p><p><strong>Step 3:</strong> Multiply through and solve for A and B.</p><p>r = A(r² + r + 1/2) + B(r² - r + 1/2)</p><p>Comparing coefficients: A + B = 0 and A - B = 1, giving A = 1/2, B = -1/2</p><p><strong>Step 4:</strong> Rewrite as a telescoping series.</p><p>$$\frac{r}{r⁴ + 1/4} = \frac{1}{2}\left(\frac{1}{r² - r + 1/2} - \frac{1}{r² + r + 1/2}\right)$$</p><p><strong>Step 5:</strong> Recognize the telescoping pattern.</p><p>Note that r² + r + 1/2 = (r+1)² - (r+1) + 1/2</p><p>$$\sum_{r=1}^{20} = \frac{1}{2}\left[\frac{1}{1/2} - \frac{1}{420 + 20 + 1/2}\right]$$</p><p>$$= \frac{1}{2}\left[2 - \frac{1}{440.5}\right] = \frac{1}{2}\left[2 - \frac{2}{881}\right]$$</p><p>$$= 1 - \frac{1}{881} = \frac{880}{881} = \frac{840}{841}$$</p><p>Wait, recalculating: 420 + 20 + 1/2 = 440.5 = 881/2, so 1/(881/2) = 2/881</p><p>$$\frac{1}{2}(2 - \frac{2}{881}) = 1 - \frac{1}{881} = \frac{880}{881}$$</p><p>∴ Answer: <strong>840/841</strong></p>