1.2.11 Determinant calculation

Problem: Calculate the determinant \( \Delta_{n} \) of size \( n \times n \). \[ \Delta_{n}=\left|\begin{array}{ccccc} x+1 & x & x & \cdots & x \\ x & x+2 & x & \cdots & x \\ x & x & x+3 & \cdots & x \\ \cdots & \cdots & \cdots & \cdots & \cdots \\ x & x & x & \cdots & x+n \end{array}\right| . \]