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quarto/precalc/exp_log_functions.qmd
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@@ -146,7 +146,7 @@ The value of `2^n` and `2.0^n` is different in `Julia`. The former remains an in
##### Example
The famous [Fibonacci](https://en.wikipedia.org/wiki/Fibonacci_number) numbers are $1,1,2,3,5,8,13,\dots$, where $F_{n+1}=F_n+F_{n-1}$. These numbers increase. To see how fast, if we *guess* that the growth is eventually exponential and assume $F_n \approx c \cdot a^n$, then our equation is approximately $ca^{n+1} = ca^n + ca^{n-1}$. Factoring out common terms gives $ca^{n-1} \cdot (a^2 - a - 1) = 0$. The term $a^{n-1}$ is always positive, so any solution would satisfy $a^2 - a -1 = 0$. The positve solution is $(1 + \sqrt{5})/2 \approx 1.618$
The famous [Fibonacci](https://en.wikipedia.org/wiki/Fibonacci_number) numbers are $1,1,2,3,5,8,13,\dots$, where $F_{n+1}=F_n+F_{n-1}$. These numbers increase. To see how fast, if we *guess* that the growth is eventually exponential and assume $F_n \approx c \cdot a^n$, then our equation is approximately $ca^{n+1} = ca^n + ca^{n-1}$. Factoring out common terms gives $ca^{n-1} \cdot (a^2 - a - 1) = 0$. The term $a^{n-1}$ is always positive, so any solution would satisfy $a^2 - a -1 = 0$. The positive solution is $(1 + \sqrt{5})/2 \approx 1.618$
That is evidence that the $F_n \approx c\cdot 1.618^n$. (See [Relation to golden ratio](https://en.wikipedia.org/wiki/Fibonacci_number#Relation_to_the_golden_ratio) for a related, but more explicit exact formula.