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?
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3See Binet's formula https://en.wikipedia.org/wiki/Fibonacci_number#Relation_to_the_golden_ratio – Donald Splutterwit Mar 09 '18 at 19:29
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4One of the suggested reasons to downvote is "This question does not show any research effort." This seems to be a prime example of that. Not only is the answer you seek found in most if not all textbooks which discuss the fibonacci sequence, it is also written very near to the top of the page in the wikipedia entry and appears several times elsewhere on this site, such as here. – JMoravitz Mar 09 '18 at 19:38
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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 $\overline{\varphi}$.
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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.
Mohammad Riazi-Kermani
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