Calculate the value of \( n \) if it is given that: \( \sum_{k=2}^{n}(5-2 k)=-\frac{800 n}{17} \)

To solve for \( n \) in the equation \[ \sum_{k=2}^{n}(5-2k)=-\frac{800 n}{17}, \] we first need to evaluate the left-hand side, which is a summation. 1. **Calculate the summation**: The expression inside the summation is \( 5 - 2k \). We can separate the summation: \[ \sum_{k=2}^{n}(5-2k) = \sum_{k=2}^{n} 5 - \sum_{k=2}^{n} 2k. \] The first part, \( \sum_{k=2}^{n} 5 \), can be calculated as follows: - The number of terms from \( k=2 \) to \( k=n \) is \( n - 1 \). - Therefore, \[ \sum_{k=2}^{n} 5 = 5(n - 1). \] The second part, \( \sum_{k=2}^{n} 2k \), can be simplified: \[ \sum_{k=2}^{n} 2k = 2 \sum_{k=2}^{n} k = 2 \left( \frac{n(n + 1)}{2} - 1 \right) = n(n + 1) - 2. \] Putting it all together, we have: \[ \sum_{k=2}^{n}(5-2k) = 5(n - 1) - (n(n + 1) - 2). \] Simplifying this expression: \[ = 5n - 5 - n^2 - n + 2 = -n^2 + 4n - 3. \] 2. **Set the summation equal to the right-hand side**: Now we set the left-hand side equal to the right-hand side: \[ -n^2 + 4n - 3 = -\frac{800 n}{17}. \] To eliminate the fraction, we can multiply through by 17: \[ 17(-n^2 + 4n - 3) = -800n. \] This simplifies to: \[ -17n^2 + 68n - 51 = -800n. \] Rearranging gives: \[ -17n^2 + 68n + 800n - 51 = 0, \] which simplifies to: \[ -17n^2 + 868n - 51 = 0. \] Multiplying through by -1 to make the leading coefficient positive: \[ 17n^2 - 868n + 51 = 0. \] 3. **Solve the quadratic equation**: We can use the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 17 \), \( b = -868 \), and \( c = 51 \): \[ n = \frac{868 \pm \sqrt{(-868)^2 - 4 \cdot 17 \cdot 51}}{2 \cdot 17}. \] First, calculate the discriminant: \[ (-868)^2 = 752624, \] \[ 4 \cdot 17 \cdot 51 = 3468, \] \[ 752624 - 3468 = 749156. \] Now, calculate \( \sqrt{749156} \): \[ \sqrt{749156} \approx 866.5. \] Now substituting back into the quadratic formula: \[ n = \frac{868 \pm 866.5}{34}. \] This gives us two potential solutions: \[ n_1 = \frac{1734.5}{34} \approx 51, \] \[ n_2 = \frac{1.5}{34} \approx 0.044. \] Since \( n \) must be a positive integer, we take \( n = 51 \). Thus, the value of \( n \) is \[ \boxed{51}. \]