The general term of the Fibonacci Sequence

The Fibonacci sequence <F_n> :0, 1, 1, 2, 3, 5, 8, 13, 21,… which satisfies F₀=0, F₁=1, and F_k+2=F_k+1+F_k. The goal of this article is to find the general form of the k^th term  F_k of  the Fibonacci sequence.

Let .

Consider , that is u_k+1= Au_k , where .

Because the characteristic polynomial of A is, the eigenvalues of A areand, and the corresponding eigenvectors are and . Of course {x₁, x₂} are linearly independent and generate u_k.

Write u₀=c₁x₁+c₂x₂, that is, and it’s easy to solve the coefficients and  . Hence

u₁=c₁Ax₁+c₂Ax₂= c₁λ₁x₁+c₂λ₂x₂ ,

 … ,

u_k=c₁A^kx₁+c₂A^kx₂= c₁λ₁^kx₁+c₂λ₂^kx₂
 .

Thus

.