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I want to talk about the Fibonacci sequence in my Linear Algebra class. So I tried to look online for examples where the sequence appears naturally.

One of the most often mentioned is that of the bees, where male bees have only a mother, and female bees have a mother and a father. Every resource I found states the above, runs through five or six generations to get 1, 1, 2, 3, 5, 8, 13, and immediately states "so, we have the Fibonacci sequence".

But that's not a proof, and I struggled to justify formally that the number of individuals in the next generation has to be the sum of the number of individuals in the previous two.

Does anyone know a formal argument to show that the number $s_n$ of individuals after $n$ generations satisfies the recursion $s_{n+1}=s_n+s_{n-1}$?

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  • $\begingroup$ General comment: I think bees are a really bad example, as a single queen bee lays thousands of eggs, but most of these eggs contain worker bees, that will not lay more eggs themselves. Thus, if you have a student slightly interested in biology or simply in bees, this example could backfire. The example I like the most makes fun of such "real world" models and starts with "Assume that rabbits are immortal." $\endgroup$ – Dirk Nov 14 '17 at 10:52
  • $\begingroup$ How to extract or reinvest money from a perfect investment. If you want to maximise income over the long term. $\endgroup$ – ctrl-alt-delor Nov 14 '17 at 17:05
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You can see it by breaking the numbers $s_n$ into parts: $s_n=f_n+m_n$, which represent the number of female and male bees, respectively, at each level of the family tree.

To find $f_{n+1}$, we note that each female bee at level $n+1$ is either the mother of a female bee at level $n$, or else the paternal grandmother of a female bee at level $n-1$. This is a one-to-one correspondence between female bees at level $n+1$ with female bees at the two previous levels, so we have $f_{n+1}=f_n+f_{n-1}$.

There is a similar argument for male bees: any male bee at level $n+1$ is either the father-in-law of a male bee at level $n$ (if his daughter's offspring is female), or else the maternal grandfather of a male bee at level $n-1$ (if his daughter's offspring is male). Thus again, $m_{n+1}=m_n+m_{n-1}$.

Finally:

$$\begin{align} s_{n+1}=f_{n+1}+m_{n+1} &= (f_n+f_{n-1})+(m_n+m_{n-1})\\ &=(f_n+m_n)+(f_{n-1}+m_{n-1})\\ &= s_n+s_{n-1} \end{align}$$

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After looking at Tony Jacobs argument, here is an argument of my own.

Start with a single male. At each generation, let $m_n$ and $f_n$ be the total number of males and females respectively, and let $s_n=m_n+f_n$. We have $$ f_0=0, f_1=1, m_0=1, m_1=0. $$ Since each male comes from a female from the previous generation, we have $$ \tag{1}m_{n+1}=f_n,\ \ \ \ n\in\mathbb N.$$ Since each female comes either from a female or a male, we have $$\tag{2} f_{n+1}=m_n+f_n,\ \ \ \ n\in\mathbb N. $$ Combining the two formulas we have $$\tag{3} f_{n+1}=m_n+f_n=f_{n-1}+f_n $$ and $$\tag{4} m_{n+1}=f_n=m_{n-1}+f_{n-1}=m_{n-1}+m_n. $$ By $(3)$ and $(4)$, $$ s_{n+1}=f_{n+1}+m_{n+1}=(f_{n-1}+f_n)+(m_{n-1}+m_n)=s_{n-1}+s_n. $$

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