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.



Positive definite matrix
If $D\_n$ is the leading principal minor of order $n$, then it seems to me
you should have
$$D\_n = 2 a\_n D\_{n-1} - a\_n^2 D\_{n-2}$$


