Understanding Fibonacci Sequence with Rabbits

recurrence-relations

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

  • $a_{n-1}$ is the total number of pairs one month ago. This includes both adults and newborns. It is how many rabbit pairs there are before the pregnant rabbits give birth for this month.
  • $a_{n-2}$ is the total number of pairs two months ago. By one month ago, they were all adults, and so all became pregnant at that time. So they are the ones giving birth now, each to one new pair. Therefore $a_{n-2}$ is also be the number of newborn pairs at this time.

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.

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