As an explorer, you have stumbled upon a remote island where you have left a pair of baby rabbits. These little creatures are adorable, but their presence on the island raises an interesting question: how will their population grow over time?
According to the problem presented, each pair of adult rabbits produces one pair of baby rabbits every month, and it takes one month for a baby rabbit to become an adult. Using this information, we can calculate the number of pairs of rabbits present on the island after n months.
After one month, the initial pair of baby rabbits will have grown into adult rabbits and produced one pair of baby rabbits. Therefore, there will be two pairs of rabbits on the island. After two months, there will be two pairs of rabbits, one of which is newborn. As the number of months increases, the population of rabbits on the island will also increase at a steady rate.
However, this problem is not just about rabbits. It is also connected to a phenomenon known as the "bee tree." The bee tree is a metaphor for exponential growth, as it illustrates how a small number of bees can quickly turn into a hive with thousands of bees. Just like the bees in the bee tree, the rabbits on the island will multiply at an exponential rate, with each pair of adult rabbits producing one pair of baby rabbits every month.
This problem serves as a reminder of how quickly a small population can grow, and the importance of understanding exponential growth. It is also a reminder of how important it is to control population growth to avoid overpopulation and depletion of resources. So next time you come across a small population of any species, remember the rabbit island and the bee tree, and think about the potential impact on the environment.
An explorer has left a pair of baby "rabbits" on an island. If baby rabbits become adults after one month, and if each pair of adult rabbits produces one pair of baby rabbits every month, how many pairs of rabbits are present after n months? (After two months there are two pairs, one of which is newborn.) Find a connection between this problem and the “bee tree” in the text.
The problem states that an explorer has left a pair of baby rabbits on an island, and each pair of adult rabbits produces one pair of baby rabbits every month. It also states that it takes one month for a baby rabbit to become an adult. Using this information, we can calculate the number of pairs of rabbits present on the island after n months.
We can use mathematical induction to solve this problem.
We can start with base case n=1, where there is one pair of baby rabbits and after 1 month, these baby rabbits will become adult and produce 1 pair of baby rabbits, so the total number of pairs will be 2.
Then, let's assume that the statement holds true for n=k, where there are f(k) pairs of rabbits after k months.
Now, we need to prove that the statement holds true for n=k+1, where there will be f(k+1) pairs of rabbits after k+1 months.
As we know from the base case, after one month, the initial pair of baby rabbits will have grown into adult rabbits and produced one pair of baby rabbits. Therefore, there will be two pairs of rabbits on the island.
We can use this information to build our solution. We can say that after k+1 months, there will be f(k) pairs of adult rabbits. These adult rabbits will produce f(k) pairs of baby rabbits in the next month. As these baby rabbits will become adult after one month, the total number of pairs will be 2*f(k) = f(k+1)
So, we have proved that the statement holds true for n=k+1, if it holds true for n=k.
Therefore, the total number of pairs of rabbits on the island after n months will be 2^n
This problem is connected to the "bee tree" phenomenon, as it illustrates how a small number of bees can quickly turn into a hive with thousands of bees. Similarly, the rabbits on the island will multiply at an exponential rate, with each pair of adult rabbits producing one pair of baby rabbits every month. The bee tree is a metaphor for exponential growth, this problem is also an example of exponential growth.
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