Fibonacci overload

The Fibonacci sequence $F_n$ is given by

$F_1 = F_2 = 1, F_{n+2} = F_{n+1} + F_{n} (n \in N)$

Prove that

$F_{2n} = \frac {F_{2n+2}^3 + F_{2n-2}^3}{9} - 2F_{2n}^3$

for all $n \geq 2$ .

#Algebra #Sharky #Fibonacci

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`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`	$\sin \theta$
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Comments

Where did you get all these proof problems? Were you inspired by a certain someone? :D

Finn Hulse - 7 years, 1 month ago

A bit, but I have been making or finding these questions through the past weeks between school assignments. Finn, one of these questions were inspired by you. Guess which one. :D

Sharky Kesa - 7 years, 1 month ago

Which one?

Finn Hulse - 7 years, 1 month ago

@Finn Hulse – A cubic game.

Sharky Kesa - 7 years, 1 month ago

@Sharky Kesa – Wait why?

Finn Hulse - 7 years, 1 month ago

@Finn Hulse – You made me think about a sequel to one of my older problems (I don't know how its related to you but when I think about a follow-up) and decided to make a sequel to A quadratic game.

Sharky Kesa - 7 years, 1 month ago

@Sharky Kesa – Oh yeah. Ironically, I got both of them wrong. :D

Finn Hulse - 7 years, 1 month ago

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}`	$\boxed{123}$