Trigonometric Functions And Equations

If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly 7 solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in \mathbf{N}\), then \(\sum_\limits{k=1}^n \frac{k}{2^k}\) is equal to: