Chapter 9.3, Problem 49E

Solution

A.

To identify: Why $a_1=1,a_2=1\text{and}{a}_3=2.$

Assume that 2 months after birth, each male-female pair of rabbits begins producing one new male-female pair of rabbits each month. Further assume that the rabbit colony begins with one newborn male-female pair of rabbits and no rabbits die for 12 months. Let represent the number of pairs of rabbits in the colony after months.

  $a_1=1$ because Assumed that the population starts with a pair male-female. first new born rabbits show up only after 2 months, thus $a_2=1$ , thus the number of pairs after 2 months is 1+1=2, so $a_3=2$ .

Given information:

  $a_1$ Is the number of pairs of rabbits after 0 months, $a_2=1$ Represents the numbers of pairs of rabbits after 1 month, $a_3$ Represents the numbers of pairs of rabbits after 2 months.

Explanation:

  $a_1$ Is the number of pairs of rabbits after 0 months, thus in the beginning. $a_1=1$ because Assumed that the population starts with a pair male-female.

  $a_2=1$ Represents the numbers of pairs of rabbits after 1 month, which is the same as in the beginning because the first new born rabbits show up only after 2 months, thus $a_2=1$ .

  $a_3$ Represents the numbers of pairs of rabbits after 2 months, when the initial pair of rabbits produces one pair of rabbits, thus the number of pairs after 2 months is 1+1=2, so $a_3=2$ .

B.

To calculate: The value of $a_4,a_5,a_6,\mathrm{......}{a}_{13}$

  $\begin{array}{c}{a}_1=a_2=1\\ a_3=2\\ a_4=2\\ a_5=2\cdot 2=4\\ a_6=4\\ a_7=2\cdot 4=8\\ a_8=8\\ a_9=2\cdot 8=16\\ a_{10}=16\\ a_{11}=2\cdot 16=32\\ a_{12}=32\\ a_{13}=2\cdot 32=64\end{array}$

Given information:

  $a_1$ Is the number of pairs of rabbits after 0 months, $a_2=1$ Represents the numbers of pairs of rabbits after 1 month, $a_3$ Represents the numbers of pairs of rabbits after 2 months.

Calculation:

Consider the given information.

  $\begin{array}{c}{a}_1=a_2=1\\ a_3=2\\ a_4=2\\ a_5=2\cdot 2=4\\ a_6=4\\ a_7=2\cdot 4=8\\ a_8=8\\ a_9=2\cdot 8=16\\ a_{10}=16\\ a_{11}=2\cdot 16=32\\ a_{12}=32\\ a_{13}=2\cdot 32=64\end{array}$ .

C.

To identify: Why the sequence , is a model for the size of the rabbit colony for a 1-year period.

  $n=1$ Represents the beginning of the year, $n=13$

represents the end of the year and the dynamics of the population is modeled by the sequence's rule of multiplication:

Given information:

  $a_1$ Is the number of pairs of rabbits after 0 months, $a_2=1$ Represents the numbers of pairs of rabbits after 1 month, $a_3$ Represents the numbers of pairs of rabbits after 2 months.

Calculation:

Consider the given information.

  $a_{2k}=a_{2k+1}=2^k$

The sequence $a_n,1\le n\le 13$ is a model for the size of the rabbit population because $n=1$ represents the beginning of the year, $n=13$

represents the end of the year and the dynamics of the population is modeled by the sequence's rule of multiplication: