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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.

Note by Khoi Nguyen Ho
3 years, 1 month ago

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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 - 3 years, 1 month ago

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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 - 3 years, 1 month ago

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

Ayanlaja Adebola - 3 years, 1 month ago

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