Fibonacci Property Refer to Example, and observe the sequence of numbers in color. Choose...

Fibonacci Property Refer to Example, and observe the sequence of numbers in color. Choose any four successive terms. Multiply the first one chosen by the fourth. Then multiply the two middle terms. Repeat this process. What do you notice when the two products are compared?

Example

EXAMPLE Solving Fibonacci’s Rabbit Problem

A man put a pair of rabbits in a cage. During the first month the rabbits produced no offspring but each month thereafter produced one new pair of rabbits. If each new pair thus produced reproduces in the same manner, how many pairs of rabbits will there be at the end of 1 year? (This problem is a famous one in the history of mathematics and first appeared in Liber Abaci, a book written by the Italian mathematician Leonardo Pisano (also known as Fibonacci) in the year 1202.)

SOLUTION

Step 1 Understand the problem. We can reword the problem as follows:

How many pairs of rabbits will the man have at the end of one year if he starts with one pair, and they reproduce this way: During the first month of life, each pair produces no new rabbits, but each month thereafter each pair produces one new pair?

Step 2 Devise a plan. Because there is a definite pattern to how the rabbits will reproduce, we can construct Table.

Table 2

Step 3 Carry out the plan. At the start of the first month, there is only one pair of rabbits. No new pairs are produced during the first month, so there is 1 + 0 = 1 pair present at the end of the first month. This pattern continues. In the table, we add the number in the first column of numbers to the number in the second column to get the number in the third.

There will be 233 pairs of rabbits at the end of one year.

Step 4 Look back and check. Go back and make sure that we have interpreted the problem correctly. Double-check the arithmetic. We have answered the question posed by the problem, so the problem is solved.