Solution
= Solution
Apply the formula to the complete quotient and use the <determinant> identity to express $b_n$. Eliminating the complete quotient from the consecutive expressions yields the stated quadratic for $q_{n+1}/q_n$. Taking the compatible signs for adjacent indices and using $c_n\le b_j$ gives
$$
\boxed{2c_na_{n+1}\le2b_na_{n+1}\le\sqrt{1-4b_nb_{n+1}}+\sqrt{1-4b_{n-1}b_n}\le2\sqrt{1-4c_n^2}.}
$$