Question

Growth of a Rabbit Colony:

A colony of rabbits begins with one pair of mature rabbits, which will produce 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?

[Hint: The Fibonacci sequence models this colony. Do you see why?]

Short answer

Number of mature rabbits after 7 months is 21.

Step-by-step solution

Step 1. Write the given information.

Two transformations that happen in one month are:

$$\begin{gathered} \mathrm{Mature}\left(M\right)\stackrel{1\mathrm{month}}{\to }\mathrm{Mature}\left(\mathrm{M}\right)+\mathrm{Immature}\left(\mathrm{I}\right) \\ \& \\ \mathrm{Immature}\left(\mathrm{I}\right)\stackrel{1\mathrm{month}}{\to }\mathrm{Mature}\left(\mathrm{M}\right) \end{gathered}$$

Step 2. Draw a tree diagram with the given information.

Using the given information, following tree diagram is obtained:

Step 3. Follow the tree diagram in Step 2 to form a series.

The number of mature pairs of rabbits from the tree diagram forms the following series:

$1,1,2,3,5,8,13,21,34,....$