0
$\begingroup$

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 ?

$\endgroup$
1
$\begingroup$

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 $$ P_n = W_n + M_n = F_n + F_{n-1} = F_{n+1} = \frac{(1 + \sqrt{5})^{n+1} - (1 - \sqrt{5})^{n+1}}{2^{n+1} \sqrt{5}}$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.