1.2.9 Determinant calculation

Problem: Calcualte the determinant of \( n \)-th order \[ \Delta_{n}=\left|\begin{array}{cccccc} a+b & a b & 0 & \ldots & 0 & 0 \\ 2 & a+b & a b & \ldots & 0 & 0 \\ 0 & 1 & a+b & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & \ldots & \ldots & a+b & a b \\ 0 & 0 & \ldots & \ldots & 1 & a+b \end{array}\right| . \]