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This can be used to calculate the $n$th Fibonacci number. \vspace{2mm}  You may notice that this formula incorporates the golden ratio, or $\phi$ (phi). There are some very special relationships between the golden ratio and the Fibonacci numbers.  \vspace{2mm}  When you take the ratio of two successive Fibonacci numbers, you get a very close approximation to the golden ratio. For example, we know that $\phi \approx 1.618...$. Consider the ratio between $2$ and $3$ which is $1.5$. Then, the ratio between $8$ and $13$ which is $1.625$.  This connection attributes to the appearance of Fibonacci numbers in nature.