Fibonacci Sequence

The Fibonacci sequence $F_{n}$ is given by

$F_{1}=F_{2}=1, F_{n+2}=F_{n+1}+F_n$ $(n\in\mathbb{N}).$

Prove that $F_{2n}=\frac{F^3_{2n+2}+F^3_{2n-2}}{9}-2F^3_{2n}$

for all $n\geq2$ .

Source: 101 Problems in Algebra book by T. Andreescu & Z. Feng

#Algebra

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$