**Sarah Hofmann**

**EMAT 6680 Summer 2006**

**Assignment 12**

**Project 4**

Leonardo Pisano Fibonacci was
a mathematician who lived in

The Fibonacci sequence is
defined as

f(0) = 1

f(1) = 1

↓

f(n+2) = f(n+1) + f(n)

We can use Excel to show the
Fibonacci numbers.

As you can see, the Fibonacci
number increases quite quickly. But it
might be interesting to look at the ratio of one Fibonacci number to the prior
Fibonacci number.

We can see that this sequence
comes to a limit very quickly. Why would
this be true? Because f(n+1) = f(n) +
f(n-1) and

f(n)/f(n+1)

f(n)/[f(n) + f(n-1)]

So each number is directly
related to the previous number, so the ratio will level out. We can think of this as f(n) = f(n-1)+f(n-2)
is to f(n-1) as f(n-1) is to f(n-2), which is the golden ratio phi = (1 +
\sqrt(5))/2.

Now we’ll look at the ratio
of every second term.

As we expected, the
difference between every third value of the Fibonacci sequence is 1 plus the
golden ratio, because we know that every element of the Fibonacci sequence is
related to the previous elements.

All that the limit of these
sequences seems to depend on is the fact that a value is directly related to
the previous two values by f(n) = f(n-1) + f(n-2) as f(n-1) is related to
f(n-2). If this is true, then the
starting values of the Fibonacci sequence should not effect the limit of the
ratios. Let’s try a few different
starting pairs and see if this is true.

As we can see from the above
three cases, the ratio is the golden ratio no matter what two initial values are
chosen, as we anticipated.