Fibonacci Proof

• Feb 4th 2011, 04:58 PM
veronicak5678
Fibonacci Proof
Prove the following identity:
(F(n+1))^2 - (F(n))^2 = F(n-1)F(n+2)
I am trying to do this with induction, and I am stuck!
• Feb 4th 2011, 11:15 PM
Drexel28
Quote:

Originally Posted by veronicak5678
Prove the following identity:
(F(n+1))^2 - (F(n))^2 = F(n-1)F(n+2)
I am trying to do this with induction, and I am stuck!

Merely note that $\displaystyle F_{n+1}^2-F_n^2=\left(F_{n+1}-F_n\right)\left(F_{n+1}+F_n\right)$. Noting then that $\displaystyle F_{n+2}=F_{n+1}+F_{n}$ and $\displaystyle F_{n+1}=F_n+F_{n-1}$ gives the desired result.
• Feb 5th 2011, 02:47 AM
Quote:

Originally Posted by veronicak5678
Prove the following identity:
(F(n+1))^2 - (F(n))^2 = F(n-1)F(n+2)
I am trying to do this with induction, and I am stuck!

Alternatively (using induction to practice! given the clarity of Drexel's response) if

$\displaystyle \left(F_{n+1}\right)^2-\left(F_n\right)^2=F_{n-1}F_{n+2}$

then we require that

$\displaystyle \left(F_{n+2}\right)^2-\left(F_{n+1}\right)^2=F_nF_{n+3}$

Proof

$\displaystyle F_nF_{n+3}=F_n\left(F_{n+1}+F_{n+2}\right)=F_nF_{n +1}+F_nF_{n+2}$

Therefore, is

$\displaystyle \left(F_{n+2}\right)^2-\left(F_{n+1}\right)^2=F_nF_{n+1}+F_nF_{n+2}\;\;?$

$\displaystyle \left(F_{n+2}\right)^2-F_nF_{n+2}=\left(F_{n+1}\right)^2+F_nF_{n+1}\;\;?$

$\displaystyle F_{n+2}\left[F_{n+2}-F_n\right]=F_{n+1}\left[F_{n+1}+F_n\right]\;\;?$

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

gives

$\displaystyle F_{n+2}F_{n+1}=F_{n+1}F_{n+2}\;\;?$

hence the hypothesis is true