Need help w/ Fibonnaci and Golden Ratio proof

[billy2908]

Let F_n and F_n+1 be successive Fibonnaci numbers. Then |(F_n+1)/(F_n) - Phi | < 1/(2(F_n)^2)

Where Phi is the Golden ratio.

 

[Eynstone]

Just use the formula for nth fibonacci number.

 

[billy2908]

It's not that simple. I tried fracturing it many times but every time it just seem to leave to the same result.

 

[Matt Benesi]

Remember that the nth Fibonacci number is:

$$F_n = \frac {1} {\sqrt{5}} \left [ \phi^n - \left ( \frac {-1}{\phi}\right)^n\right]$$

and you can multiply F_n² through your equation:

$$\left | F_{n+1} \times F_{n} - \phi \times F^2_n \right | < \frac {1}{2}$$

What happens when you set the whole equation to terms of $\phi$?

Note that for the first 3 Fibonacci numbers: 0,1,1 the formula doesn't work. Additionally, as we get deeper into the Fibonacci sequence, $F_n$ as $n \to \infty$, what does $\left | F_{n+1} \times F_{n} - \phi \times F^2_n \right |$ approach? hint: you might see the value first for a specific very low n

 

[blue_raver22]

Sure, I would be happy to help with your proof for the Fibonacci and Golden Ratio relationship. First, let's define the Fibonacci sequence as a series of numbers where each number is the sum of the two previous numbers, starting with 0 and 1. So, the first few numbers in the sequence would be 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.

Now, let's look at the relationship between two successive Fibonacci numbers, F_n and F_n+1. We can express this relationship as:

F_n+1 = F_n + F_n-1

We can also express the Golden Ratio, Phi, as:

Phi = (1 + √5)/2

Now, let's substitute F_n+1 and F_n into the first equation:

F_n+1 = F_n + F_n-1

F_n+1/F_n = (F_n + F_n-1)/F_n

F_n+1/F_n = 1 + F_n-1/F_n

Now, we can use the definition of the Golden Ratio, Phi, and substitute it into the equation:

F_n+1/F_n = 1 + F_n-1/F_n

F_n+1/F_n = 1 + Phi/F_n

Next, we can rearrange the equation to isolate the term for Phi:

F_n+1/F_n - 1 = Phi/F_n

Finally, we can take the absolute value of both sides of the equation to get:

|F_n+1/F_n - 1| = |Phi/F_n|

Now, we can use the fact that F_n is always a positive integer to simplify the equation further:

|F_n+1/F_n - 1| = |Phi/F_n|

|F_n+1/F_n - 1| = |Phi|/F_n

Since Phi is a constant value, we can substitute it with its numerical value:

|F_n+1/F_n - 1| = 0.618/F_n

Now, let's look at the right side of the equation, where we have 1/(2(F_n)^2). We can rewrite this as:

1/(2(F_n)^2) = 0.5/F_n^2

So, now we have the following equation:

|F_n+1/F_n -