We have $a_1,a_2,...,a_n\in (0,1)$ and matrix M= \begin{bmatrix}2a_1&a_2&a_3&.&.\\a_2&2a_2&a_3&.&.\\a_3&a_3&2a_3&.&.\\.&.&.&.&.\end{bmatrix}

We need to check if M is positive definite.

I am trying to evaluate it's determinant as a polynomial in $a_i$ as principal minor are of the same type. And using that frame a condition for positive definiteness of M.