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0.2 Integers and Fibonacci Numbers and Ratios of Successive Same

Whoop de doo!

Let's try something more interesting. Suppose we put 1 in box B3 and in B4 put '=B2+B3'. Now copy B4 into B5...B1000
What will appear are what are called the Fibonacci numbers.

So?

What are their properties? They have lots and lots of them. We can observe that they grow without limit. But suppose we look at the ratio of two successive Fibonacci numbers. To do this put in C5='B5/b4' (case doesn't matter) and copy this down column C. What do you see?

What is that number?

The Fibonacci numbers and the positive integers as well each form an infinite sequence; in column C you see what is called a convergent sequence.

To figure out what the number you see is, imagine that we had exactly f ( n ) = r f ( n − 1 ) for some r for many consecutive n values. What can we say about r ?

Since the defining property of the Fibonacci numbers is f ( n ) = f ( n − 1 ) + f ( n − 2 ) , we could then write r 2 f ( n − 2 ) = r f ( n − 2 ) + f ( n − 2 ) , which we can solve for r .

The larger solution is called the "Golden Ratio", and is the number you see.

By the way, the Fibonacci numbers count the number of different ways of inserting n dominoes into a 2 by n grid, so that each domino covers two adjacent boxes.

Exercises:

0.1 Set this all up on your own machine. Solution

0.2 Prove that the Fibonacci numbers count the number of different ways of inserting n dominoes into a 2 by n grid, so that each domino covers two adjacent boxes. Solution

0.3 Make a definition of convergence of a sequence that reflects the property that you see. Solution

0.4 This procedure produces a solution to the quadratic equation indicated above. Given any quadratic with integer coefficients, we can produce a recursion as above and by substituting it into B4 and copying it down, look at what happens to it. Try doing this with some quadratics, and find another for which we get a solution, and one which we don't. What happens with the cubic x 3 = x + 1 ? Solution

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