Binet's Formula
Binet's formula is an explicit formula used to find the th term of the Fibonacci sequence.
It is so named because it was derived by mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre.
Formula
If is the
th Fibonacci number, then
.
Proof
To derive a general formula for the Fibonacci numbers, we can look at the interesting quadraticBegin by noting that the roots of this quadratic are
TOTO SLOT according to the quadratic formula. This quadratic can also be written as
From this, we can write expressions for all
:
We note that
Let the roots of our original quadratic be
and
Since both
and
are roots of the quadratic, they must both satisfy
So
and
Subtracting the second equation from the first equation yields
This yields the general form for the nth Fibonacci number:
Proof using Recursion
The Fibonacci recursive relation is This is a constant coefficient linear homogenous recurrence relation. We also know that
and
Thus, its characteristic equation is
which has solutions
Let
and
We get that
Plugging in our initial conditions, we get
Since subtracting
from
we get
Since
and
Therefore,
Therefore, the general form of the
th Fibonacci number is
~peelybonehead