# Fibonacci Sequence The Fibonnaci Sequence is a number sequence in which the $n$-th term is defined as a sum of the previous two terms. $ F_n = \begin{cases} n=1:& 1 \\ n=2:& 1 \\ n>2:& F_{n-1} + F_{n-2} \end{cases} $ Any term in the Fibonnaci Sequence can be calculated using Binet's Formula: $ F_n=\frac{\left(\frac{1+\sqrt 5}{2} \right)^n-\left(\frac{1-\sqrt 5}{2} \right)^n}{\sqrt 5} $ The ratio of two consecutive terms approaches [[The Golden Ratio]] as $n \rightarrow \infty$. $ \lim_{n \rightarrow \infty} \left( \frac{F_{n+1}}{F_n} \right) = \frac{1+\sqrt 5}{2} = \varphi $