<p>The value of <br>\((n-1)(n-\omega)(n-\omega^2)\)<br> where the series \(S_{20}\), \(S_{10}\), \(S_{31}\), \(S_{15}\) are computed as shown. Which of the following values are correct?</p><p>\(S_{20} = \frac{(20^2)(19^2)}{4} - 20 + 1\)</p><p>\(S_{10} = \frac{(10^2)(9^2)}{4} - 10 + 1\)</p><p>\(S_{31} = \frac{(31^2)(30^2)}{4} - 31 + 1\)</p><p>\(S_{15} = \frac{(15^2)(14^2)}{4} - 15 + 1\)</p>
<p>\(S_{20} = 36081\)</p>
<p>\(S_{10} = 2016\)</p>
<p>\(S_{31} = 216195\)</p>
<p>\(S_{15} = 11011\)</p>
Step-by-Step Solution
Key Concept: Recognize that S_n follows the pattern S_n = [n²(n-1)²]/4 - n + 1, which can be rewritten as S_n = [n(n-1)]²/4 - n + 1. Verify each given expression matches this general formula by substituting the specific values of n.
<p><strong>Step 1:</strong> Identify the general formula for S_n: S_n = [n²(n-1)²]/4 - n + 1 = [n(n-1)]²/4 - n + 1</p><p><strong>Step 2:</strong> Verify S₂₀: [20²·19²]/4 - 20 + 1 = [400·361]/4 - 19 = 36,100 - 19 = 36,081 ✓</p><p><strong>Step 3:</strong> Verify S₁₀: [10²·9²]/4 - 10 + 1 = [100·81]/4 - 9 = 2,025 - 9 = 2,016 ✓</p><p><strong>Step 4:</strong> Verify S₃₁: [31²·30²]/4 - 31 + 1 = [961·900]/4 - 30 = 216,225 - 30 = 216,195 ✓</p><p><strong>Step 5:</strong> Verify S₁₅: [15²·14²]/4 - 15 + 1 = [225·196]/4 - 14 = 11,025 - 14 = 11,011 ✓</p><p>∴ Answer: A, B, C, D (All four expressions are correct)</p>
Correct Answer: A, B, C, D