Assume that each man in generation $n-1$ will make one woman survive to the next generation, and each woman in generation $n-1$ will give birth one man and one woman to the next gernation. Assume that the zero generation has only one man. What is the size of population in the generation $n$.
$\textbf{Sol.}$ So generation $0$ : $1$ m, gen $1$ : $1$ w, gen $2$ : $1$ m and $1$ w, gen $3$: $1$ m and $2$ w, gen $4$ : $2$ m and $3$ w, ...
Let $M_n, W_n$ denote the number of man and woman in generation $n$. Let $P_n$ donote then total population in $n$ generation.
Then $$M_n = W_{n-1}, W_n = M_{n-1} + W_{n-1}$$ with $M_0 = 1, W_0 = 0$ and $$P_n = W_n + M_n.$$
Is there a way to have a formula for $P_n$ ?
Usually, for example like fibonacci sequence, we can set $f_n = r^n$ to solve the formula, but this one is like two variables, how to do it ?
You have $$W_n = M_{n-1} + W_{n-1} = W_{n-2} + W_{n-1}$$ with $W_0=0, W_1=1$. This is Fibonacci number, so you can write $$W_n = F_n$$ The $M_n$ is then simply $$M_n = W_{n-1} = F_{n-1}$$ So overall you've got $$ P_n = W_n + M_n = F_n + F_{n-1} = F_{n+1} = \frac{(1 + \sqrt{5})^{n+1} - (1 - \sqrt{5})^{n+1}}{2^{n+1} \sqrt{5}}$$