Fibonacci ??? Help me

Have anyone noticed that when we take 4 consecutive numbers in the Fibonacci list: f(x), f(x+1), f(x+2), f(x+3). We will have:

f(x) * f(x+3) - f(x+1) * f(x+2) =1 or -1

For example: 2 * 8 - 3 * 5 =1

p/s: It is just my view, I do not sure if it is always true.

#Algebra #NumberTheory #FibonacciNumbers #Ilovenumbertheory #Fibonacci

Easy Math Editor

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Comments

That is a great observation.

Let me suggest a way of continuing:
Can you list out the values of $x$ where the expression is 1, and the values of $x$ where the expression is -1?
Do you know Binet's formula which gives you the value of $f(x)$ ?

Calvin Lin Staff - 6 years, 7 months ago

Hmm.........AM SPEECHLESS

Ayanlaja Adebola - 6 years, 7 months ago

oh yeah, just adding x, x+1, x+2, x+3 into the Binet's and then minus two products, I found that f(x) * f(x+3) - f(x+1) * f(x+2)= -1 * (-1)^x. Great. Thank you.

Khoi Nguyen Ho - 6 years, 7 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}`	$\boxed{123}$