Binet's Formula

February 19, 2026

math

fibonacci

Binet’s formula is a closed-form expression for the nnth number of the Fibonacci sequence, giving FnF_n directly without computing all earlier terms.

Fn=φn−ψnφ−ψ=(1+52)n−(1−52)n5F_n=\frac{\varphi^n-\psi^n}{\varphi-\psi}=\frac{\left(\frac{1+\sqrt5}{2}\right)^n-\left(\frac{1-\sqrt5}{2}\right)^n}{\sqrt5}

Here, φ\varphi is the golden ratio and ψ\psi is its conjugate.

φ=1+52≈1.61803…,ψ=1−52≈−0.61803…\varphi=\frac{1+\sqrt5}{2}\approx 1.61803\ldots,\quad \psi=\frac{1-\sqrt5}{2}\approx -0.61803\ldots

To derive this expression, we can start with the Fibonacci rule.

F0=0,F1=1,Fn=Fn−1+Fn−2.F_0=0,\quad F_1=1,\quad F_n=F_{n-1}+F_{n-2}.

Because the recurrence has constant coefficients, we try an exponential form Fn=rnF_n=r^n, since shifting nn just multiplies by rr. Now the recurrence turns into an algebra equation for rr.

rn=rn−1+rn−2r^n=r^{n-1}+r^{n-2}

Factor out and cancel rn−2r^{n-2}, given rn−2≠0r^{n-2}\neq 0:

rn−2(r2)=rn−2(r+1)r2=r+1r2−r−1=0.\begin{aligned} &r^{n-2}(r^2)=r^{n-2}(r+1) \\ &r^2=r+1 \\ &r^2-r-1=0. \end{aligned}

Solving for the roots of r2−r−1=0r^2-r-1=0 gives us 1±52\frac{1\pm\sqrt5}{2}, the golden ratio and its conjugate:

φ=1+52,ψ=1−52\varphi=\frac{1+\sqrt5}{2},\qquad \psi=\frac{1-\sqrt5}{2}

At this point, we know we have two basic sequences that each obey Xn=Xn−1+Xn−2X_n=X_{n-1}+X_{n-2}: namely Xn=φnX_n=\varphi^n and Xn=ψnX_n=\psi^n. Because the rule is just adding previous terms, any weighted sum of two valid sequences is also valid.

So the most general sequence that follows the Fibonnaci rule has the form Xn=Aφn+BψnX_n = A\varphi^n + B\psi^n. We can choose A,BA,B so that X0=0X_0=0 and X1=1X_1=1 to model the Fibonacci sequence.

Given F0=0F_0 = 0:

0=Aφ0+Bψ0=A+B⇒B=−A0=A\varphi^0+B\psi^0=A+B \Rightarrow B=-A

Given F1=1F_1 = 1:

1=Aφ+Bψ=Aφ−Aψ=A(φ−ψ)1=A\varphi+B\psi=A\varphi-A\psi=A(\varphi-\psi)

Given φ−ψ=5\varphi-\psi=\sqrt5:

A=15,B=−15A=\frac1{\sqrt5},\quad B=-\frac1{\sqrt5}

Substituting the values for AA and BB, we get the closed-form expression for the Fibonnaci sequence:

Fn=Aφn+Bψn=15 φn−15 ψn=φn−ψn5=(1+52)n−(1−52)n5\begin{aligned} F_n &= A\varphi^n + B\psi^n \\ &= \frac{1}{\sqrt5}\,\varphi^n - \frac{1}{\sqrt5}\,\psi^n \\ &= \frac{\varphi^n-\psi^n}{\sqrt{5}} \\ &=\frac{\left(\frac{1+\sqrt5}{2}\right)^n-\left(\frac{1-\sqrt5}{2}\right)^n}{\sqrt5} \end{aligned}