Proving Fibonacci Sequence and Golden Ratio

[matrix_204]

I was just working on proving the Fibonacci sequence and the Golden ratio i think, but i was having one problem.
I was asked to prove that An=Bn, where An+2=An+1 + An (the fibonacci sequence) and that Bn=1/root5 [(1+root5/2)^n - (1 - root5/2)^n].
I understood the whole problem and was half way in completing until i wasn't able to go any further. Knowin that the limit existed i came up with
L=[1+/- root 5 ]/2
then i was told that C+D=1 and C(L)+D(-L)=1, and by solving for C and D by these two equations, i would get
Bn=1/root5 [(1+root5/2)^n - (1 - root5/2)^n].

But the values of C and D cancel out or i m getting like a zero for one of them. What did i do wrong, and what am i suppose to do next?

 

[marlon]

I was just working on proving the Fibonacci sequence and the Golden ratio i think, but i was having one problem.
I was asked to prove that An=Bn, where An+2=An+1 + An (the fibonacci sequence) and that Bn=1/root5 [(1+root5/2)^n - (1 - root5/2)^n].
I understood the whole problem and was half way in completing until i wasn't able to go any further. Knowin that the limit existed i came up with
L=[1+/- root 5 ]/2
then i was told that C+D=1 and C(L)+D(-L)=1, and by solving for C and D by these two equations, i would get
Bn=1/root5 [(1+root5/2)^n - (1 - root5/2)^n].

But the values of C and D cancel out or i m getting like a zero for one of them. What did i do wrong, and what am i suppose to do next?

Hmmm,...i am not sure what you mean. Is this Bn :

$$\frac{1}{\sqrt{5}} \left\{(1+\sqrt{\frac{5}{2}})^n - (1-\sqrt{\frac{5}{2}})^n \right\}$$


Then you probably need to show whether Bn exhibits the caracteristics of a Fibonacci-series right?

marlon

 

[matrix_204]

no bn=1/root5[((1 + root5)/2)^n - ((1 - root5)/2)^n]

 

[matrix_204]

$$Bn =\frac{1}{\sqrt{5}} \left\{(\frac{1+\sqrt{5}}{2})^n - (\frac{1-\sqrt{5}}{2})^n \right\}$$