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May 12th, 2009 - Sarah's Blog [entries|archive|friends|userinfo]
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May 12th, 2009

Idiot's guide? [May. 12th, 2009|10:40 pm]
Sarah
Someone was explaining the golden ratio on Digg.
I know a bit of math(s) - enough to get by with programming - but I never had a teacher who'd go slow enough for dumb old me to keep up - and out of school I was always too fascinated by shiny things to try and find a book at the library. (This was before the internets!)

I'd love to be able to understand the following, are there any good online links that would help me?

I agree, maths is fun - but for me, lots of it is impenetrable, but when I do find out how an equation 'works', I'm all "woohoo!".



This is because the closed form of the fibonacci sequence is
f(n) = 1/(sqrt(5) * ( (1+sqrt(5)/2)^(n+1) - ( (1-sqrt(5)/2)^(n+1)

This would be when f(0)=f(1)=1, not when f(0)=0 and f(1)=1, which would just be changing the exponent to n, as opposed to n+1.

Not that anyone cares, but a brief derivation.

f(n)=f(n-1)+f(n-2)
f(n)-f(n-1)+f(n-2)=0

Now we assume there exists a closed form such that a(n)=f(n), and a(n)=c^n

c^n+c(n-1)+c(n-2)=0
c^(n-2)*(c^2-c-1)=0

The solutions to c^2-c-1=0, are (1+sqrt(5))/2, (1-sqrt(5))/2. (yes the golden ratio)

With which we can say, due to it being a second order recurrence relation.

f(n)=A*((1+sqrt(5))/2)^n + B*((1-sqrt(5))/2)^n

Then it's just a matter of solving the equation

Math is fun!
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