Chapter 12.1, Problem 85AYU

Growth of a Rabbit Colony A colony of rabbits begins with one pair of mature rabbits, which produces a pair of offspring (one male, one female) each month. Assume that all rabbits mature in $1$ month and produce a pair of offspring (one male, one female) after $2$ months. If no rabbits ever die, how many pairs of mature rabbits are there after $7$ months? See figure. top right. (Hint: A Fibonacci sequence models this colony. Do you see why?)

To determine

The number of pairs of mature rabbits after $7$ months if no rabbits ever died. Where, a colony of rabbits begins with one pair of mature rabbits, which produces a pair of offspring (one male, one female) each month. Assume that all rabbits mature in $1$ month and produce pair of offspring (one male, one female) after $2$ months.

Answer to Problem 85AYU

Solution:

There will be $21$ pairs after $7$ months if no rabbit ever died.

Explanation of Solution

Given information:

A colony of rabbits begins with one pair of mature rabbits, which produces a pair of offspring (one male, one female) each month. Assume that all rabbits mature in $1$ month and produce pair of offspring (one male, one female) after $2$ months.

Explanation:

By observing the provided figure, In the first month, there is only one pair of mature rabbits.

In the second month, there is one pair of mature rabbits and one pair of offspring.

In the third month, there are two pairs of mature rabbit and one pair of offspring.

In the fourth month, there are three mature pairs with two pairs of offsprings.

Thus, in each next month the number of pairs is just the addition of pairs in the previous two months.

This information is represented by the recursive sequence

$a_n=a_{n-2}+a_{n-1}$ where $a_n$ is number of mature pairs after ${(n-1)}^{th}$ month.

Here, $a_1=1,a_2=1$ and $a_3=a_1+a_2=1+1=2$.

Similarly, $a_4=a_2+a_3=1+2=3$

$a_5=a_3+a_4=2+3=5$

$a_6=a_4+a_5=3+5=8$

$a_7=a_5+a_6=5+8=13$

$a_8=a_6+a_7=8+13=21$

Therefore, there will be $21$ pairs of mature rabbits after $7$ months.