1.2.12 Determinant calculation

Problem: Calculate the determinant of the order \( (n+1) \times(n+1) \). \[ \Delta_{n+1}=\left|\begin{array}{cccccc} 0 & 1 & 1 & \cdots & 1 & 1 \\ 1 & 0 & a_{1}+a_{2} & \cdots & a_{1}+a_{n-1} & a_{1}+a_{n} \\ 1 & a_{2}+a_{1} & 0 & \cdots & a_{2}+a_{n-1} & a_{2}+a_{n} \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 1 & a_{n-1}+a_{1} & a_{n-1}+a_{2} & \cdots & 0 & a_{n-1}+a_{n} \\ 1 & a_{n}+a_{1} & a_{n}+a_{2} & \cdots & a_{n}+a_{n-1} & 0 \end{array}\right| . \]