<p>Find the value of \(\displaystyle\sum_{r=1}^{n} \left[(n+1)r - r^2\right]\) when \(n = 15\).</p>
Step-by-Step Solution
Key Concept: Separate the summation into two parts using linearity of summation, then apply standard formulas for ∑r and ∑r². This converts a complex expression into manageable algebraic sums.
<p><strong>Step 1:</strong> Separate the summation using linearity:</p><p>$$\sum_{r=1}^{n} [(n+1)r - r^2] = (n+1)\sum_{r=1}^{n} r - \sum_{r=1}^{n} r^2$$</p><p><strong>Step 2:</strong> Apply standard formulas with n = 15:</p><p>$$\sum_{r=1}^{15} r = \frac{15 \times 16}{2} = 120$$</p><p>$$\sum_{r=1}^{15} r^2 = \frac{15 \times 16 \times 31}{6} = \frac{7440}{6} = 1240$$</p><p><strong>Step 3:</strong> Substitute (n+1) = 16:</p><p>$$= 16 \times 120 - 1240 = 1920 - 1240 = 680$$</p><p>∴ Answer: <strong>680</strong></p>
Correct Answer: 680