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.
If is the th Fibonacci number, then .
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 according to the quadratic formula. This quadratic can also be written as From this, we can write expressions for all : We note thatLet the roots of our original quadratic be and Since both and are roots of the quadratic, they must both satisfy SoandSubtracting 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