I am trying to figure out the best way to prove a formula quickly because there is a chance it could be on exam with a lot of questions and on average we should spend two minutes per question. My tools for this problem have been the cofactor expansion theorem mainly but I have been starting to think there might be a way to apply induction although I have not been able to isolate the argument after going over computing the determinant for $n=3, n=4$. My guess is induction is the fastest way to prove the following formula.
$ A = \begin{pmatrix} 0 & 1 & 1 & 1 & \ldots & 1 \\ 1 & 0 & a_1+a_2 & a_1+a_3 & \ldots & a_1+a_n\\ 1 & a_2+a_1 & 0 & a_2+a_3 & \ldots & a_2 +a_n \\ 1 & a_3+a_1 & a_3 + a_2 & 0 & \ldots & a_3 +a_n \\ . & . & . & .& . &.\\ 1 & a_n+a_1 & a_n+a_2 & a_n+a_3 & \ldots & 0 \end{pmatrix} $
How do you show $\det(A) = (-1)^n 2^{n-1} \sum\limits_{i=1}^{n} a_1 a_2 \ldots a_{i-1} a_{i+1} \ldots a_n$