(8) \( \sum_{r=2}^{m} 2(3-r)>-800 \)

To solve the inequality: \[ \sum_{r=2}^{m} 2(3 - r) > -800 \] let's break it down step by step. ### Step 1: Simplify the Sum First, factor out the constant \(2\): \[ \sum_{r=2}^{m} 2(3 - r) = 2 \sum_{r=2}^{m} (3 - r) \] Now, evaluate the sum inside: \[ \sum_{r=2}^{m} (3 - r) = \sum_{r=2}^{m} 3 - \sum_{r=2}^{m} r \] Calculate each part separately: 1. **Sum of Constants:** \[ \sum_{r=2}^{m} 3 = 3(m - 1) \] 2. **Sum of \(r\) from 2 to \(m\):** \[ \sum_{r=2}^{m} r = \sum_{r=1}^{m} r - 1 = \frac{m(m + 1)}{2} - 1 \] Putting it all together: \[ \sum_{r=2}^{m} (3 - r) = 3(m - 1) - \left(\frac{m(m + 1)}{2} - 1\right) \] Simplify: \[ = 3m - 3 - \frac{m^2 + m}{2} + 1 \] \[ = \frac{6m - 6 - m^2 - m + 2}{2} \] \[ = \frac{-m^2 + 5m - 4}{2} \] Multiply by 2 (from the earlier factor): \[ 2 \sum_{r=2}^{m} (3 - r) = -m^2 + 5m - 4 \] ### Step 2: Set Up the Inequality Now, set the simplified sum greater than \(-800\): \[ -m^2 + 5m - 4 > -800 \] Add \(800\) to both sides: \[ -m^2 + 5m + 796 > 0 \] Multiply both sides by \(-1\) (remember to reverse the inequality sign): \[ m^2 - 5m - 796 < 0 \] ### Step 3: Solve the Quadratic Inequality Find the roots of the quadratic equation \(m^2 - 5m - 796 = 0\) using the quadratic formula: \[ m = \frac{5 \pm \sqrt{25 + 3184}}{2} = \frac{5 \pm \sqrt{3209}}{2} \approx \frac{5 \pm 56.66}{2} \] This gives two approximate roots: \[ m \approx \frac{5 + 56.66}{2} \approx 30.83 \quad \text{and} \quad m \approx \frac{5 - 56.66}{2} \approx -25.83 \] Since \(m\) represents the upper limit of a sum starting from \(r = 2\), it must be a positive integer. Therefore, the valid range for \(m\) is: \[ -25.83 < m < 30.83 \] Considering \(m\) must be an integer: \[ m \leq 30 \] ### Final Answer For the inequality to hold, \(m\) must be at most 30.