Fibonacci sequence proof

May 2010
1
0
Hi,

I'm having trouble solving the following proof on my study guide for an exam:

Prove for the Fibonacci sequence F_k


F_k*F_k - F_(k-1)*F_(k-1) = F_k*F_(k+1) - F_(k+1)*F_(k-1)

for k>=1



Could anyone show me what I need to be doing? Any help would be appreciated. Thanks!
 
Oct 2008
1,116
431
\(\displaystyle F_k+F_{k-1}=F_{k+1}\)

Multiplying both sides of the equation by \(\displaystyle F_k-F_{k-1}\),

\(\displaystyle (F_k+F_{k-1})(F_k-F_{k-1})=F_{k+1}(F_k-F_{k-1})\)

\(\displaystyle F_k^2-F_{k-1}^2=F_kF_{k+1}-F_{k+1}F_{k-1}\)

\(\displaystyle F_kF_k-F_{k-1}F_{k-1}=F_kF_{k+1}-F_{k+1}F_{k-1}\)
 
Dec 2009
3,120
1,342
Hi,

I'm having trouble solving the following proof on my study guide for an exam:

Prove for the Fibonacci sequence F_k


F_k*F_k - F_(k-1)*F_(k-1) = F_k*F_(k+1) - F_(k+1)*F_(k-1)

for k>=1



Could anyone show me what I need to be doing? Any help would be appreciated. Thanks!
Alternatively, the Fibonacci recursion relation is

\(\displaystyle F_{k+1}=F_k+F_{k-1}\)

We are asked to prove \(\displaystyle F_kF_k-F_{k-1}F_{k-1}=F_kF_{k+1}-F_{k+1}F_{k-1}\)

This contains \(\displaystyle F_{k-1}\) on the left, hence \(\displaystyle F_{k-1}=F_{k+1}-F_k\)

Therefore, rewriting the left side

\(\displaystyle F_kF_k-F_{k-1}F_{k-1}=F_kF_k-\left(F_{k+1}-F_k\right)\left(F_{k+1}-F_k\right)\)

\(\displaystyle =F_kF_k-\left(F_{k+1}F_{k+1}-2F_{k+1}F_k+F_kF_k\right)\)

\(\displaystyle =2F_kF_{k+1}-F_{k+1}F_{k+1}=F_kF_{k+1}+F_kF_{k+1}-F_{k+1}F_{k+1}\)

\(\displaystyle F_{k+1}\) is a factor of the last 2 terms (it's a factor of all 3 of course), giving

\(\displaystyle F_kF_{k+1}+F_{k+1}\left(F_k-F_{k+1}\right)=F_kF_{k+1}-F_{k+1}\left(F_{k+1}-F_k\right)\)

In brackets is \(\displaystyle F_{k-1}\)
 
Oct 2009
769
87
Letting you know

Just picked up a book on Fibonacci numbers by Alfred Posamentier (2007) which is interesting reading.