\( \sum _ { i = 4 } ^ { m } ( 2 i + 8 ) > 10500 \)

To solve the inequality \( \sum _ { i = 4 } ^ { m } ( 2 i + 8 ) > 10500 \), we first need to compute the sum. The expression \( 2i + 8 \) can be simplified as follows: \[ \sum _ { i = 4 } ^ { m } ( 2 i + 8 ) = \sum _ { i = 4 } ^ { m } 2i + \sum _ { i = 4 } ^ { m } 8 \] Calculating each part: 1. The sum of \(2i\): \[ \sum _ { i = 4 } ^ { m } 2i = 2 \sum _ { i = 4 } ^ { m } i = 2 \left( \frac{m(m+1)}{2} - \frac{3(4)}{2} \right) = m(m+1) - 24 \] 2. The sum of \(8\): \[ \sum _ { i = 4 } ^ { m } 8 = 8(m - 4 + 1) = 8(m - 3) = 8m - 24 \] Combining these results: \[ \sum _ { i = 4 } ^ { m } ( 2 i + 8 ) = m(m+1) - 24 + 8m - 24 = m^2 + 9m - 48 \] Now, we set the inequality: \[ m^2 + 9m - 48 > 10500 \] Rearranging gives: \[ m^2 + 9m - 10548 > 0 \] Next, we can find the roots of the equation \(m^2 + 9m - 10548 = 0\) using the quadratic formula: \[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-9 \pm \sqrt{9^2 - 4(1)(-10548)}}{2(1)} \] Calculating the discriminant: \[ 9^2 + 4 \times 10548 = 81 + 42192 = 42273 \] Now, putting this into the formula: \[ m = \frac{-9 \pm \sqrt{42273}}{2} \] Calculating \( \sqrt{42273} \approx 205.56\): \[ m = \frac{-9 \pm 205.56}{2} \] This gives us two solutions: \[ m_1 \approx \frac{196.56}{2} \approx 98.28 \quad \text{and} \quad m_2 \approx \frac{-214.56}{2} \approx -107.28 \] Since \(m\) must be a positive integer, we take the ceiling of \(m_1 \approx 98.28\), so \(m\) must be greater than or equal to 99. Thus, the final solution is \(m \geq 99\).