hereโs the quadratic formula
i think itโs hard
fuck its hard
what is f
yes
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Donโt google the quartic formula.
I will do it
Doesnโt look too hard to me
The mysticism is kinda cool, but itโs not really that weird and unexpected if you look at it right.
The definition of the golden ratio is that two positive numbers, a and b are in a โgolden ratioโ if a/b = (a+b)/a.
This is equivalent to saying that a/b = 1 + b/a, so now we substitute a/b = ๐, and get:
๐ = 1/๐ + 1
๐ - 1/๐ - 1 = 0, now multiply both sides by ๐
๐2 - ๐ - 1 = 0
So ๐ is just the positive solution to the equation, which is (1+โ5)/2.
The reason it appears in the fibonacci sequence is a bit more complicated if you want to understand why that is, or easier if you want to just take my word for it because I really didnโt like this part of last years classes and had to open my textbook to find this stuff so Iโm too lazy to properly prove everything.
Kinda long and you have to take a lot for granted
Itโs a case of solving a difference equation, meaning finding an explicit formula for the nth element of a series if youโre given a recursive definition and starting elements.
For the fibonacci sequence the recursive definition is fn+2 = fn+1 + fn, and f1 = f2 = 1.
You can start from 0 or index differently, but this is simplest for the explanation.
How you solve a second order difference equation like this one, where an elements value depends on 2 previous ones, is by first finding a so-called โcharacteristic equationโ.
When the recursive definition is in the form an+2 = pan+1 + qan, like this one, the equation is:
t2 = pt + q, where t is a parameter that will be needed for the final solution.
Notice how t is the solution to the equation t2 - t - 1 = 0, similar to how ๐ is the positive solution to ๐2 - ๐ - 1 = 0. However, t can be both of the solutions, and that plays a part in solving the difference equation.
Once you find that t1 = (1+โ5)/2, t2 = (1-โ5)/2, the solution to the difference equation is (again without proof):
fn = C1t1n + C2t2n
You have to find C1 and C2 in relation to the given starting elements, in this case f1 = f2 = 1, by just substituting t1 and t2 into the recursive definition and solving.
You get that C1 = 1/โ5 and C2 = -1/โ5.
The explicit formula for the nth element of the fibonacci sequence is:
fn = 1/โ5 * [((1+โ5)/2)n - ((1-โ5)/2)n]
To show that the ratio of two elements approaches ๐, find the limit of fn+1 / fn as n approaches infinity, which is best shown with an image because of the big fraction:
The ((1-โ5)/2)n parts go to 0 because (1-โ5)/2 is between -1 and 0, and the 1/โ5 parts immediately cancel out, so what remains is equal to ๐.
For the last part, itโs clear why thatโs the case when you view ๐ as the positive solution to ๐2 - ๐ - 1 = 0.
๐2 is obviously just ๐ + 1
๐3 is the same as ๐(๐ + 1) which is ๐2 + 1 which is ๐ + 2
๐4 is (๐ + 1)2 which is ๐2 + ๐ + 1 which is 2๐ + 2
And so on, and so on
For the other way around, when the exponent is negative
๐-1 is 1/๐ which is (๐2 - ๐) / ๐ which is ๐ - 1
You get the idea
Sorry if I ruined it for you. Youโre welcome if this made it more interesting.
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what the fuck
f(x) means the function of x, basically the fancy (and more useful) way of writing y (e.g. y = x + 2)
this guy is talking enchant table language
wow and now I have a migraine
I remember someone on the forums telling me that โno formula should face discrimination, except the quartic formulaโ
Wait, it was you lol