Find a recurrence relation for the number of pairs of rabbits after n months if (1)
initially there is one pair of rabbits who were just born, and (2) every month each
pair of rabbits that are over one month old have a pair of offspring (a male and a
female)
I know that this is a fibonacci sequence where the recurrence relation is
$a_n=a_{n-1}+a_{n-2}$
but I'm a little confused on something.
In this situation,
$a_{n-1}$ represents the pairs of rabbits that are grown up/old/fertile correct?
and
$a_{n-2}$ represents the pairs of rabbits that are just born/young correct?
thus, you would sum the two together to find the total amount of pairs of rabbits correct?
I test this with
$a_0=1$ which represent a pair rabbits that were just born/are young
$a_1=1$ which represent those pair of rabbits growing up/ready for breeding
so from that I think that I understand what $a_{n-1}$ and $a_{n-2}$ represent but I want to make sure I do by checking with this site.
Do I have a good understanding of what $a_{n-1}$ and $a_{n-2}$ represent?
Best Answer
For each $n, a_n$ is the total number of pairs of rabbits after $n$ months. So
Thus the number of adult rabbit pairs at this point in time is $a_{n-1}$ and the number of newborn rabbit pairs at this point of time is $a_{n-2}$. Since every rabbit pair is either adult or newborn, that is all of the rabbits.