For any natural $n$ prove that
\begin{gather*} {\frac { q \left|\begin {array}{ccc} {q}^{n+2}& \left( {\frac {p}{q}} \right) ^{n+2}& \left( {p}^{-1} \right) ^{n +2}\\ {q}& {\frac {p}{q}} & {p}^{-1} \\ 1&1&1 \end {array} \right| - p \left|\begin {array}{ccc} {q}^{n+1}& \left( {\frac {p}{q}} \right) ^{n+1}& \left( {p}^{-1} \right) ^{n +1}\\ {q}& {\frac {p}{q}} & {p}^{-1} \\ 1&1&1 \end {array} \right| }{ \left| \begin {array}{ccc} {q}^{2}&{\frac {{p}^{2}}{{q}^{2}}}&{p}^{-2} \\ q&{\frac {p}{q}}&{p}^{-1}\\ 1&1 &1\end {array} \right| }}\\|| \\q\frac{\begin{vmatrix} q^{n+1} & p^{-(n+1)} \\ 1 & 1 \end{vmatrix}}{\begin{vmatrix} q& p^{-1} \\ 1 & 1 \end{vmatrix}}. \end{gather*} I can prove it by brute force expansion but I hope there are more elegant way to do it.
See also My old similar question