<p>The value of \(\displaystyle\sum_{r=16}^{30}(r+2)(r-3)\) is equal to</p>
Step-by-Step Solution
Key Concept: Expand (r+2)(r-3) = r² - r - 6, then use standard summation formulas for ∑r² and ∑r to compute the sum from r=16 to 30.
<p><strong>Step 1:</strong> Expand the product: (r+2)(r-3) = r² - 3r + 2r - 6 = r² - r - 6</p><p><strong>Step 2:</strong> Split the sum: ∑_{r=16}^{30}(r² - r - 6) = ∑_{r=16}^{30}r² - ∑_{r=16}^{30}r - ∑_{r=16}^{30}6</p><p><strong>Step 3:</strong> Convert using ∑_{r=a}^{b} = ∑_{r=1}^{b} - ∑_{r=1}^{a-1}:</p><p>∑_{r=16}^{30}r² = ∑_{r=1}^{30}r² - ∑_{r=1}^{15}r² = [30·31·61/6] - [15·16·31/6] = 9455 - 1240 = 8215</p><p>∑_{r=16}^{30}r = ∑_{r=1}^{30}r - ∑_{r=1}^{15}r = [30·31/2] - [15·16/2] = 465 - 120 = 345</p><p>∑_{r=16}^{30}6 = 6(30-16+1) = 6(15) = 90</p><p><strong>Step 4:</strong> Combine: 8215 - 345 - 90 = 7780</p><p>∴ Answer: <strong>7780</strong></p>
Correct Answer: A