**4.Generate a Fibonnaci sequence
in the first column using **

**
f(0)=1, f(1)=1, **

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

** **

a. Construct the ratio of each pair of adjacent terms in the
Fibonnaci sequence.

What
happens as n increase?

Using
excel we construct the
Fibonnaci sequence and found some interesting points.

As n increases, the ratio of each pair
of adjacent terms became approached one number.

The spreadsheet expressed it to 0.618933É
We can say this convergence to one number.

What about the ratio of every second term?

I could find the ratio of every second term also became approached one number 1.

In the ratio of the second term is faster converged than the one of the each pair of adjacent terms.

Excel
file for Fibonnaci sequence

b. Explore sequences where f(0) and f(1) are some
arbitrary integers
other than 1.

If f(0)=1 and f(1)=3, then your sequence
is a Lucas Sequence. All such sequences,

however,
have the same limit of the ration of successive terms.

I constructed a Lucas Sequence and
investigate the result. Like the
Fibonnaci sequence,

the
ratio of each pair of adjacent terms in it converged the same number in the
Fibonnaci sequence.

I investigated the ratio of every second
term and found that it had the same limit

of
the second termÕs ratio of the
Fibonnaci sequence. Moreover, in the Lucas Sequence,

second
termÕs ratio converge to 1 faster than the Fibonnaci sequence.

I had great interest that the convergence speed was faster as initial number was increased.

So I tried to make a another sequence: f(0)=1 and f(1)=4. For my expectation

a result was similar to other sequences. The ratio of successive terms had the same limit

of the Fibonnaci and Lucas sequence, and the ratio of second term also converged to 1.

However, in the last experiment, the speed of convergence was not faster than a Lucas sequence.