Question

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).

Answer 1

Answer:

Relation : $\frac{F_{N}}{F_{N-1} }$

Ratio reaches limit at value : 1.61803398

Step-by-step explanation:

To solve this, we need to use the Fibonacci Sequence.

Let F(n) be the number of pair of rabbits after n months.

At 0 month , we have 0 pair of rabbit.

Initially we have 1 pair of rabbit that was just born so , at 1 month , we have 1 pair of rabbit

F(1) = 1

And next month another pair of rabbit is born so:

F(2) = 1

As given, each pair of rabbit gives birth to another pair of rabbit after a month so :

After 3 months, the rabbits will be as follows:

Initial pair ----> Gives 1 pair

2nd pair -----> Gives 1 pair

New pair of rabbits now = 1+1 = 2

So to write this in terms of n, we can simply write it as :

F(n) = F(n-1) +F(n-2)

Which gives the value of rabbits after 'n' months

                                                                                                             

We can also do this using successive term ratio which gives us the recurrence relation for pairs of rabbits after n months:

for n month, the ratio will be equal to $\frac{F_{N}}{F_{N-1} }$

Using Excel, and by starting at month 2 to avoid dividing by zero, we see that ratio reaches limit of 1.61803398