MHB Solve Fibonacci Squares: Formula for Difference of Squares

[pleasehelpsos]

I need help with this problem...

By experimenting with numerous examples in search of a pattern, determine a simple formula for (Fn+1)2−(Fn−1)2; that is, a formula for the difference of the squares of two Fibonacci numbers.What does this question want? What is it asking for?

 

[Greg]

Hi pleasehelpsos and welcome to MHB! :D

...(Fn+1)2−(Fn−1)2...

By the above, did you intend

$$F_{n+1}^2-F_{n-1}^2$$

?

 

[MarkFL]

I think I would begin with:

$$F_{n+1}=F_{n}+F_{n-1}$$

What do you get when you square both sides?

 

[pleasehelpsos]

How do I pick which sequence number it goes along with?

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Hi pleasehelpsos and welcome to MHB! :D
By the above, did you intend

$$F_{n+1}^2-F_{n-1}^2$$

?

Yes

 

[Greg]

The Fibonacci sequence defined by $F_n=F_{n-1}+F_{n-2}$ has initial terms $0,1,1,2,3,5,8,13,21...$ Pick the 1st and 3rd and apply the rule, the 2nd and 4th and apply the rule, the 3rd and 5th and apply the rule and so on. Then look for a pattern in the results.

 

[pleasehelpsos]

I honestly still do not understand

 

[pleasehelpsos]

like i got zero when i had tried to do n=1 n=2 n=3 and n=4

for the first one i tried to sub in f1 but i don't know what i am doing

 

[Greg]

Okay; let's do some more terms of the Fibonacci sequence:

$$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...$$

Apply the rule to them:

$$1^2-0^2=1,2^2-1^2=3,3^2-1^2=8,5^2-2^2=21,8^2-3^2=55,13^2-5^2=169-25=144...$$

Now look at those results and at the Fibonacci sequence above them. What do you notice?

 

[MarkFL]

Consider the closed form for the $n$th Fibonacci number:

$$F_n=\frac{\varphi^n-(-\varphi)^{-n}}{\sqrt{5}}$$ where $$\varphi=\frac{1+\sqrt{5}}{2}$$

Hence:

$$F_n^2=\frac{\varphi^{2n}-2(-1)^n+(-\varphi)^{-2n}}{5}$$

And so:

$$F_{n+1}^2-F_{n-1}^2=\frac{\varphi^{2(n+1)}-2(-1)^{n+1}+(-\varphi)^{-2(n+1)}}{5}-\frac{\varphi^{2(n-1)}-2(-1)^{n-1}+(-\varphi)^{-2(n-1)}}{5}$$

$$F_{n+1}^2-F_{n-1}^2=\frac{\varphi^{2n}\left(\varphi^2-\varphi^{-2}\right)+(-\varphi)^{-2n}\left((-\varphi)^{-2}-(-\varphi)^{2}\right)}{5}$$

Observing that:

$$\varphi^2-\varphi^{-2}=\sqrt{5}$$

$$(-\varphi)^{-2}-(-\varphi)^{2}=-\sqrt{5}$$

We have:

$$F_{n+1}^2-F_{n-1}^2=\frac{\varphi^{2n}-(-\varphi)^{-2n}}{\sqrt{5}}=F_{2n}$$