**We wish to create a spreadsheet composed
of the fibonacci numbers and then investigate various ratios of
these numbers. In the first column of our spreadsheet we create
the fibonacci numbers,that is those numbers beginning with 1 and
1 where each subsequent number is the sum of the previous two
numbers. This is easily done through the use of formulas in the
spreadsheet to create as many of these numbers as we desire .**

**The second column of numbers is generated
by the ratio of adjacent numbers in the first column. This produces
an interesting result in that these ratios approximate a well
known number called the golden ratio. Greeks were aware of this
ratio of numbers as being particularly architecturally pleasing
to the eye. We see this ratio exemplified in one of the most well
known Greek antiquities, the Parthenon.**

**The third column is generated by the ration
of every other entry in the first column. This produces a constant
number which is the Golden ratio plus one.**

**An interesting question to ask is whether
these ratios are particular to the fibonacci sequence or might
other numbers work just as well in the first column. Lets try
an example:**

**This is interesting. The same ratios are
generated. This result begs the question as to whether the same
ratio is generated by any two beginning numbers. Lets generalize
slowly and start by investigating any two numbers such that the
second number is 4 more than the first number. This would produce
a spreadsheet which looks like this:**

**Careful inspection of these ratios in column
b shows that the numerical part of the expression is exactly the
fibonacci sequence and the fractional algebraic part of the expression
is approaching 0 for all x. (The denominator is increasing in
size quickly while the numerator is decreasing.) So it appears
that for any two numbers one of which is 4 greater than the other,
which generate a sequence formed by the addition of the previous
two numbers, the golden ratio is approximated. Let try a positive
and negative number just to further convince ourselves:**

**Or we will also try two negative numbers.
The results are the same. We may generate a further generalization
spreadsheet by choosing the first two numbers x and x+n. The result
is closely akin to the generalized result above where the second
number is 4 greater than any number x. I leave the production
of that spreadsheet to your enjoyment.**