General term for fibonacci sequence

Question

Can we define the general term for the Fibonacci Sequence? I mean we can make a generating function for fibonacci sequence which equals $\dfrac1{1-x-x^2}$, but is there any way to find the general term of fibonacci sequence?

Answer 1

Yes, it is possible. Let $$ \varphi=\frac{1+\sqrt{5}}{2} \ \text{ and }\overline{\varphi}=\frac{1-\sqrt{5}}{2} $$ Then,

$$ F_n=\frac{\varphi^n-\overline{\varphi}^n}{\sqrt{5}} $$ for all $n \in \mathbb{N}$

For information, the number $\varphi$ is called the golden ratio. This formula can be shown simply by searching for geometric sequence $F_n=F_0r^n$ solution of the Fibonacci's recursive formula $$ F_{n+2}=F_{n+1}+F_n \Leftrightarrow \left(r^2-r-1\right)F_n=0 $$ The solution of $ \ r \mapsto r^2+r+1$ are $\varphi$ and $\overline{\varphi}$.

Answer 2

Consider the matrix $$A=\begin {bmatrix} 1&1\\1&0\end {bmatrix}$$

Note that $$A^n=\begin {bmatrix} F_{n+1}&F_{n} \\ F_{n} &F_{n-1} \end {bmatrix}$$ Where $F_n$ is the $n_{th}$ Fibonacci's number.