"Start with a pair of rabbits, (one male and one female) born on January 1. Assume that all months are of equal length and that :

1. rabbits begin to produce young two months after their own birth;

2. after reaching the age of two months, each pair produces a mixed pair, (one male, one female), and then another mixed pair each month thereafter; and

3. no rabbit dies.

How many pairs of rabbits will there be after one year?"

Fibonacci's Solution: The Fibonacci Sequence!

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

So, at the end of the year, there will be 144 pairs of rabbits, all resulting from the one original pair born on January 1 of that year.

~~~~~~Question~~~~~~~~~

Keep the assumptions of rabbit reproduction the same as for the Fibonacci numbers, but with one change. Now assume that the time to fertility for a newborn is 2 months rather than 1. Starting with a newborn male and female, make a table which gives the number of pairs of rabbits after each month for 10 months. What is the mathematical relationship that gives the terms of this sequence of numbers?

72

2,048

To find the number of pairs of rabbits after each month for 10 months, we can create a table as follows:

Month | Number of Pairs of Rabbits
-------------------------------
1 | 1 (the initial pair)
2 | 0 (no reproduction yet)
3 | 1 (the initial pair becomes fertile)
4 | 1 (the initial pair reproduces)
5 | 2 (the initial pair reproduces again)
6 | 3 (the initial pair and the first offspring reproduce)
7 | 4 (the initial pair, first and second offspring reproduce)
8 | 7 (the initial pair, first, second, and third offspring reproduce)
9 | 12 (the initial pair, first, second, third, and fourth offspring reproduce)
10 | 20 (the initial pair, first, second, third, fourth, and fifth offspring reproduce)

From the table, we can observe that the number of pairs of rabbits follows a pattern. The sequence can be described by a mathematical relationship:

Number of pairs of rabbits = (Number of pairs of rabbits in the previous month) + (Number of pairs of rabbits two months ago)

Based on this relationship, we can calculate the number of pairs of rabbits for any given month beyond the 10 months shown in the table.

To solve this question, we need to create a table that shows the number of pairs of rabbits after each month for 10 months. Let's go step by step:

Month 1: In the first month, we start with one pair of rabbits (one male and one female).

Month 2: The pair of rabbits is not yet fertile, so no new pairs are born.

Month 3: The original pair becomes fertile, and they give birth to one more pair of rabbits (one male and one female). So now we have 2 pairs of rabbits.

Month 4: The original pair gives birth to another pair, and the previous pair becomes fertile and gives birth to one more pair. So now we have 3 pairs of rabbits.

Month 5: The original pair gives birth to one more pair, the pair from the previous month becomes fertile and gives birth to another pair, and the first-born pair becomes fertile and gives birth to one more pair. So now we have 5 pairs of rabbits.

Month 6: Following the same pattern, the number of pairs of rabbits will be 8.

Month 7: 13 pairs of rabbits.

Month 8: 21 pairs of rabbits.

Month 9: 34 pairs of rabbits.

Month 10: 55 pairs of rabbits.

By observing the pattern, we can see that this sequence of numbers follows the Fibonacci sequence, however, shifted by one. The Fibonacci sequence starts with 1, 1, 2, 3, 5... while our sequence starts with 1, 2, 3, 5, 8...

To summarize, the mathematical relationship that gives the terms of this sequence of numbers is a shifted Fibonacci sequence. The nth term can be found by adding the (n-1)th term and the (n-2)th term.