# Fibonacci sequence help

#### 06Rousher

##### New member
Problem is: "By experimenting with numerous examples in search of a pattern, determine a simple formula for (F n+1)^2-(F n-1)^2; That is, a formula for the difference of the squares of two Fibonacci numbers."

The n+1 and n-1 should be smaller by the F but I dont know how to do that on a computer

Any help is appreciated

#### MarkFL

Staff member
We are asked to find a formula for:

$$\displaystyle A_{n}=F_{n+1}^2-F_{n-1}^2$$

So, as suggested, let's see if a pattern develops:

$$\displaystyle A_1=F_2^2-F_0^2=1^2-0^2=1=F_2$$

$$\displaystyle A_2=F_3^2-F_1^2=2^2-1^2=3=F_4$$

$$\displaystyle A_3=F_4^2-F_2^2=3^2-1^2=8=F_6$$

$$\displaystyle A_4=F_5^2-F_3^2=5^2-2^2=21=F_8$$

At this point, we could state the induction hypothesis $P_n$:

$$\displaystyle F_{n+1}^2-F_{n-1}^2=F_{2n}$$

Can you proceed?

#### 06Rousher

##### New member
Proceed with continuing the pattern?

Im not understanding the hypothesis of F 2n aswell