# Thread: Real Analysis: Fibonacci Numbers

1. ## Real Analysis: Fibonacci Numbers

Here is the problem:

2. Originally Posted by Phyxius117
Here is the problem:

I will give you a push.

I leave the base case to you.

For our inductive step, assume that $\displaystyle F_iF_{i - 2} = (F_{i - 1})^2 + (-1)^i$ (1).

We show that $\displaystyle F_{i + 1}F_{i - 1} = (F_i)^2 + (-1)^{i + 1}$.

Note that

$\displaystyle F_{i + 1}F_{i - 1} = (F_i + F_{i - 1})F_{i - 1}$

$\displaystyle = F_i F_{i - 1} + (F_{i - 1})^2$

Now, change $\displaystyle (F_{i - 1})^2$ using the equation in (1) and simplify to get the desired result.

Try it

3. ty so much!! will try it!