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General Fibonacci sequence

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Post Thu Dec 08, 2016 12:56 pm

Posts: 98
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:
\[\begin{array}{l}
G_n^2 - {G_{n - 1}}{G_{n + 1}} = {\left( { - 1} \right)^{n - 1}}, \\
{G_m}{G_n} + {G_{m + 1}}{G_{n + 1}} = {G_{m + n + 1}}, \\
G_n^2 + G_{n + 1}^2 = {G_{2n + 1}}, \\
G_{n + 1}^2 - G_{n - 1}^2 = a{G_{2n}}. \\
\end{array}\]

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