Here was my previous question: Proof by Induction Using Fibonacci numbers

There was a similar one in existence already over here: Inductive proof of a formula for Fibonacci numbers

The one that was answered already had a decent answer but it isn't as detailed as I wish it could be.

hypergeometric mentioned this in his/her answer:

"Note also that 1+1/ϕ=ϕ and 1−ϕ=−1/ϕ."

However, I don't know where this was obtained. Everything else I can see, but this is not immediately clear to me. Any help would be appreciated.

  • $\begingroup$ You know that $\phi$ satisfies $\phi^2 = \phi+1$, so you just have to divide each side by $\phi$. $\endgroup$ – Kevin Quirin Oct 2 '15 at 7:50

It is clear that $$1+\frac{1}{\phi}=1+\frac{\sqrt{5}-1}{2}=\frac{\sqrt{5}+1}{2}=\phi$$.

Similarly $$1-\phi=1-\frac{\sqrt{5}+1}{2}=-\frac{\sqrt{5}-1}{2}=-\frac{1}{\phi}$$.

  • $\begingroup$ Yes, now I see it. I did the math wrong originally when I tried going through with the induction myself. $\endgroup$ – NotAStudentForReal Oct 2 '15 at 8:05

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