\begin{align} D_{n}&= \begin{vmatrix} 1+a_1&a_2&\cdots&a_{n-1}&a_{n}\\ a_1&1+a_2&\cdots&a_{n-1}&a_{n}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ a_1&a_2&\cdots&1+a_{n-1}&a_{n}\\ a_1&a_2&\cdots&a_{n-1}&1+a_{n} \end{vmatrix}\\ &\xlongequal{\underset{n\ge k\ge 2}{r_{k}-r_{k-1}}} \begin{vmatrix} 1+a_1&a_2&\cdots&a_{n-1}&a_{n}\\ -1&1&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&0\\ 0&0&\cdots&-1&1 \end{vmatrix}\\ &\xlongequal{\underset{n\ge k\ge 2}{c_{k-1}+c_{k}}} \begin{vmatrix} 1+\sum_{k=1}^n\limits a_k&\sum_{k=2}^n\limits a_k&\cdots&a_{n-1}+a_n&a_{n}\\ 0&1&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&0\\ 0&0&\cdots&0&1 \end{vmatrix}\\ &=1+\sum_{k=1}^n\limits a_k. \end{align}
