need help about sequence of fibonacci

if you cant see or that picture seem not clear , please feel free to ask me, thank you

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Comments

The generating function for the Fibonacci numbers is $G(x) = \frac{x}{1-x-x^2}$ . (Think of the infinite geometric series and what terms combine to make $x^n$ ...). So, you want to solve $\frac{a}{1-a-a^2} = 2$ or $2a^2 +3a -2 = 0$ . Since we must have $0<a<\frac{\sqrt{5}-1}{2}$ , this leaves $a=\frac{1}{2}$ .

Eric Edwards - 7 years, 10 months ago

Could you elaborate on how you found that closed form?

Tim Vermeulen - 7 years, 10 months ago

Here is an interesting idea. Try to follow these steps carefully:

$G(x) = x+x^2+2 x^3+3 x^4+5 x^5+8 x^6+13 x^7+\text{...}$

$\frac{G(x)}{x} = 1+x+2 x^2+3 x^3+5 x^4+8 x^5+13 x^6+\text{...}$

$\frac{G(x)}{x}-G(x) = 1+x^2+x^3+2 x^4+3 x^5+5 x^6+8 x^7+\text{...}$

$x G(x) = x^2+x^3+2 x^4+3 x^5+5 x^6+8 x^7+13 x^8+\text{...}$

And finally,

$\frac{G(x)}{x}-G(x)-x G(x)=1$

Solve this for $G(x)$ to get $G(x) = \frac{x}{1-x-x^2}$

Ivan Stošić - 7 years, 10 months ago

@Ivan Stošić – Wow, that's amazing. It's like the first time I saw a proof of the sum of a geometric series, where suddenly all the terms canceled out.

Tim Vermeulen - 7 years, 10 months ago

I highly recommend Wilf's Generatingfunctionology, which deals with formal generating functions. The second edition is available online for free on the author's site. Using this, you can actually generate a formal proof.

George Williams - 7 years, 10 months ago

@George Williams – that's very nice book, thanks

NurFitri Hartina - 7 years, 10 months ago

@George Williams – That's looking really good, thank you.

Tim Vermeulen - 7 years, 10 months 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}`	$\boxed{123}$