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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 isf(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)=0Now we assume there exists a closed form such that a(n)=f(n), and a(n)=c^nc^n+c(n-1)+c(n-2)=0c^(n-2)*(c^2-c-1)=0The 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)^nThen it's just a matter of solving the equationMath is fun! link post comment 