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*What is
a spreadsheet?*

A spreadsheet is software used to facilitate the manipulation of large amounts of data. For example, if one wanted to sort the phone book by last name, first name, address, or phone number, it could be accomplished by inputing the data into a spreadsheet and using the sort command by perference. There are many commands in spreadsheets that can expedite data entry, as well.

Let's consider the ramifications of this software on mathematics by exploring a couple of well-known number sequences. For these explorations, we will use Microsoft Excel.

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**Fibonnaci Sequence**

This number sequence looks like this:

The first two numbers in the sequence
can be entered into a spreadsheet and the infinite numbers that
follow can be found using a simple formula in the subsequent cells
of the spreadsheet. *Do we need to manually enter the formula
in each subsequent cell?* NO! We can simply enter the formula,
"=A1+A2", into A3 and fill the entire column with this
formula (A is the column and 1, 2, 3, ... is the row) :

Viola...the beginning of the Fibonnaci sequence! Now, let's look at the ratio of each pair of adjacent terms in the Fibonnaci sequence:

*What happens as n increases?* Well, we see the ratio reaching a number
of approximately 1.618 -- the golden ratio!

Now, *what about the ratio of
every second term, third term, etc.?*

Each column, starting with B, represents the ratio of adjacent terms, every other term, third term, fourth term, fifth term, etc. Two observations from this investigation are: 1) the Fibonnaci sequence continues diagonally as the first ratio number, and 2) the Fibonnaci formula is discovered by comparing the addition sequence of the ratios by row (i.e. notice how the last row of ratios compare to the row underlined below it) -- the two underlined rows match!!!

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**Lucas Sequence**

This number sequence has the same formula as the Fibonnaci sequence; however, f(0) = 1 and f(1) = 3:

Notice how the ratio reaches a number of approximately 1.618 -- the golden ratio, again!!!