Question

How could I prove that this gives the nth Fibonacci number?

Answer 1

\[\LARGE F(n)= \frac{2 \frac{d^n}{dn^n}[\sinh(n \ln \Phi)]}{\sqrt{5} \ln ^n(\Phi)}\]

Answer 2

Phi is the golden ratio.

Answer 3

*

Answer 4

MY GOOOOOOOOD !!

@Kainui what is this man ? O.o

May be we can use mathematical induction ?

Answer 5

Now this is an answer ;)

\(\color{blue}{\text{Originally Posted by}}\) @ganeshie8
*
\(\color{blue}{\text{End of Quote}}\)

Answer 6

I am pretty sure it is right, I just came up with it by a clever technique. =P

Wait for your proof don't forget: \[\LARGE * \square\]

Answer 7

Yeah see ^^^^^ I told him that too :P

Answer 8

Ok should I give hints or give the answer or let you think or search the internet more?

Answer 9

inteeeeeeeeernnnnnnnet :P

Answer 10

Well I have to go for dinner now..... Bye :)

Answer 11

Ok good luck =P

Answer 12

For dinner ? O.o ?

Answer 13

sinh(x) = sin(ix)?

Answer 14

That's a step in the somewhat right direction dan.

Answer 15

Although it's sinh(x)=-i sin(ix)

http://www.wolframalpha.com/input/?i=sinh(x)%3D-isin(ix)&t=crmtb01

Answer 16

okay

Answer 17

You should consider the exponential definition of the hyperbolic trig functions.

Answer 18

yeah

Answer 19

e^ix=cosx+isinx
e^-ix=cosx-isinx
sinx=(e^ix+e^-ix)/2i
-isin(ix)=-(e^-x+e^x)/2
therefore
sinhx=(e^x-e^-x)/2

Answer 20

Haha yeah that's one way to do it.

Answer 21

now i feel like there is some binomial expansion tricking coming up or some taylor series expansion

Answer 22

Well what do you know about the derivatives of hyperbolic sine and cosine?

Answer 23

but first ln stuff

Answer 24

umm cant we get it from -isin(ix)

Answer 25

d/dx(sinhx)=d/dx(-isin(ix))=-cos(ix)=coshx

Answer 26

cos(ix)*

Answer 27

I think you're over complicating it a little bit but you seem to be doing alright.

Answer 28

n ln phi = (ln phi )^n

Answer 29

\[\LARGE \frac{d}{dx}(\sinh(x))= \frac{d}{dx}(\frac{e^x-e^{-x}}{2})\]\[\LARGE \frac{d}{dx}(\frac{e^x-e^{-x}}{2})=\frac{e^x+e^{-x}}{2}=\cosh(x)\]

Answer 30

stop using equation editor

Answer 31

why cant u be a noob like me

Answer 32

okay so our bracket will simplify nicely so far im not sure if that is helpful yet

Answer 33

wait n ln phi is not (ln phi)^n it is ln(phi^n)!

THERE NO EQUATION EDITOR HAH!

Answer 34

oh really

Answer 35

Wow I like that, it makes it look more confusing if I make it have ln(phi^n) in the sinh on top and ln^n(phi) on bottom. Very mysterious looking haha.

Answer 36

i think we are trying to prove F(n) = f(n-1) + f(n-2)