The general Fibonacci sequence is defined by

\[{G_{n + 2}} = a{G_{n + 1}} + {G_n},\;n \ge 0,\;\;{G_0} = 0,\;{G_1} = 1.\]

Prove the following results:

\[\begin{array}{l}

G_n^2 - {G_{n - 1}}{G_{n + 1}} = {\left( { - 1} \right)^{n - 1}}, \\

{G_m}{G_n} + {G_{m + 1}}{G_{n + 1}} = {G_{m + n + 1}}, \\

G_n^2 + G_{n + 1}^2 = {G_{2n + 1}}, \\

G_{n + 1}^2 - G_{n - 1}^2 = a{G_{2n}}. \\

\end{array}\]