I've never felt dumber: me understand Fibonacci identity.

[moe darklight]

This is crazy. I have no idea what the textbook is saying at the end.

if we have E(0)=A, E(1)=B, And E(n)=E(n-1)+E(n-2); then E(n)=F(n-1)A+F(n)B

(Where F(n) denotes a Fibonacci number.)

So far, so good. Then this flies at me out of nowhere:

We can start with two consecutive Fibonacci numbers A=F(a) and B=F(a+1). Then the sequence E(n) is just the Fibonacci sequence shifted to the left (agreed). Hence we get the following identity:

F(a+b+1)=F(a+1)F(b+1)+F(a)F(b)

We do?? Where the hell did that come from?

I've never stared at something for so long without having the slightest clue what is going on. I held up the whole class today for like 20 minutes, and it seemed to make sense when the prof explained it . . . but now I look at it again, and whatever I learned, I un-leaned. Ugh, I'm so frustrated.

 

[hamster143]

E(n)=F(n-1)A+F(n)B

E(n)=F(n-1)E(0)+F(n)E(1)

E(n+1)=F(n)E(0)+F(n+1)E(1)

and since E(n)=F(b+n) (by definition),

F(b+n+1)=F(n)F(b)+F(n+1)F(b+1)

 

[moe darklight]

of course. Thanks.