# Fibonacci sequence proof

#### mattgavin

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!

#### alexmahone

$$\displaystyle F_k+F_{k-1}=F_{k+1}$$

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

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

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

$$\displaystyle F_kF_k-F_{k-1}F_{k-1}=F_kF_{k+1}-F_{k+1}F_{k-1}$$

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 $$\displaystyle F_{k-1}$$

#### wonderboy1953

Letting you know

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