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Fibonacci sequence help

06Rousher

New member
Feb 8, 2014
9
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

Administrator
Staff member
Feb 24, 2012
13,775
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
Feb 8, 2014
9
Proceed with continuing the pattern?

Im not understanding the hypothesis of F 2n aswell
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Proceed with continuing the pattern?

Im not understanding the hypothesis of F 2n aswell
I mean can you continue the proof by induction. The hypothesis is what we notice appears to be the pattern that arises when computing the first several terms of the sequence we are asked to explore. Have you been using induction in your course?
 

06Rousher

New member
Feb 8, 2014
9
No i have no clue on induction

Im helping a friend with his work and trying to understand it myself cause I know it will be in my future. So I havent had guidelines or someone to teach me, just been trying to do this on my own