1.2.7 Determinant calculation

Problem: Calculate the determinant of the \( n \)-th order \[ \Delta=\left|\begin{array}{ccccccc} 1 & 2 & 3 & \ldots & \ldots & n-1 & n \\ 1 & 3 & 3 & 4 & \ldots & n-1 & n \\ 1 & 2 & 5 & 4 & \ldots & n-1 & n \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 2 & 3 & \ldots & \ldots & 2 n-3 & n \\ 1 & 2 & 3 & \ldots & \ldots & n-1 & 2 n-1 \end{array}\right| . \]