||[May. 12th, 2009|10:40 pm]
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.
Now we assume there exists a closed form such that a(n)=f(n), and a(n)=c^n
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!