# Thread: Fibonacci members and powers of the golden ratio

1. ## Fibonacci members and powers of the golden ratio

Hello. I searched for a topic for the same question, but my queries were fruitless, although they returned some similar questions. If i have missed it, please lock this one, and redirect me to the proper thread.
It's not a homework assignment, but a personal inquiry due to some home studies I'm performing. I have a little background in maths, but haven't done anything of the sorts in quite a while ( > 4 years), and was looking for some confirmation on my work. Anyhow, here's the problem.
Prove that $i$th Fibonacci number satisfies the equality
$
F_n = \frac{{\phi}^n - {\hat{\phi}}^n}{\sqrt{5}}
$

Where $\phi$ is the golden ratio and $\hat{\phi}$ is its conjugate.

What I've done:
Proof by induction.
Base cases:
$
F_0 = {\phi}^0 - {\hat{\phi}}^0
$

$
\\F_0 = 0
$

$
F_1 = {\phi}^1 - {\hat{\phi}}^1
$

$
\\F_1 = 1
$

Inductive hypothesis:
Suppose that it holds for [LaTeX ERROR: Compile failed]
The same for $\hat{\phi}$.

From that we trivially arrive at
$
\frac{{\phi}^{n+1} - {\hat{\phi}}^{n+1}}{\sqrt5} = \frac{{\phi}^n + {\phi}^{n-1}}{\sqrt5} - \frac{{\hat{\phi}}^{n} + {\hat{\phi}}^{n-1}}{\sqrt5}
$

Which is what we were trying to prove. Can you please confirm my reasoning? I'm not sure if I'm allowed such an assumption in my inductive hypothesis, and also my reasoning looks a little circular. If there's a gap, can you please point it out to me, as well as any hints at getting the proper solution?

2. Originally Posted by ossinator
Hello. I searched for a topic for the same question, but my queries were fruitless, although they returned some similar questions. If i have missed it, please lock this one, and redirect me to the proper thread.
It's not a homework assignment, but a personal inquiry due to some home studies I'm performing. I have a little background in maths, but haven't done anything of the sorts in quite a while ( > 4 years), and was looking for some confirmation on my work. Anyhow, here's the problem.
Prove that $i$th Fibonacci number satisfies the equality
$
F_n = \frac{{\phi}^n - {\hat{\phi}}^n}{\sqrt{5}}
$

Where $\phi$ is the golden ratio and $\hat{\phi}$ is its conjugate.

What I've done:
Proof by induction.
Base cases:
$
F_0 = {\phi}^0 - {\hat{\phi}}^0
$

$
\\F_0 = 0
$

$
F_1 = {\phi}^1 - {\hat{\phi}}^1
$

$
\\F_1 = 1
$

Inductive hypothesis:
Suppose that it holds for $F_n$ and $F_{n-1}$
We also know, that $\phi$ and $\hat{\phi}$ are the roots of the equation $x^2 = x + 1$
Therfore, if we multiply both sides of the equation by ${\phi} ^ {n-1}$, we get ${\phi}^{n+1} = {\phi}^{n} + {\phi}^{n-1}$
The same for $\hat{\phi}$.

From that we trivially arrive at
$
\frac{{\phi}^{n+1} - {\hat{\phi}}^{n+1}}{\sqrt5} = \frac{{\phi}^n + {\phi}^{n-1}}{\sqrt5} - \frac{{\hat{\phi}}^{n} + {\hat{\phi}}^{n-1}}{\sqrt5}
$

Which is what we were trying to prove. Can you please confirm my reasoning? I'm not sure if I'm allowed such an assumption in my inductive hypothesis, and also my reasoning looks a little circular. If there's a gap, can you please point it out to me, as well as any hints at getting the proper solution?
That is a very nice proof.

3. Does that mean that it works?
Also, there's a mistake in my base cases, a $\sqrt{5}$ is missing.

4. Originally Posted by ossinator
Does that mean that it works?
Also, there's a mistake in my base cases, a $\sqrt{5}$ is missing.
Yes, you should have $F_0 = ({\phi}^0 - {\hat{\phi}}^0)/\sqrt5$ and $F_1 = ({\phi}^1 - {\hat{\phi}}^1)/\sqrt5$. But apart from that the proof is completely correct.