###### Example 2.3.1

Use a spreadsheet to generate values for the sequence defined by

Our sequence is defined explicitly by the map \(n \mapsto x_n\text{.}\) In our spreadsheet, we will use one column to define a subsequence of the values for the index. Then we will define a second column for the values of the sequence.

We will make the first column, `A`

, contain the values for the index. The first entry in the column will be a label. So type `n`

in the cell `A1`

. Our first value for the index i \(n=0\text{,}\) so type `0`

in cell `A2`

. The next value will be \(n=1\text{,}\) so we type `1`

in cell `A3`

.

Now, it would be very tedious to type all of the values for the index once entry at a time. We take advantage of technology to have the spreadsheet apply a pattern to complete the rest of the column. Select the two cells we have already created, `A2`

and `A3`

. You should see a grab box, usually in the bottom right corner. If you drag that box down the column and then let go, the spreadsheet will follow your pattern for all of the cells you select before releasing. You can make this column of index values as long as you desire, within the limits of the program you use.

We are now ready to create the column containing the values of the sequence. We start with a label for the column in `B1`

, for example typing `x_n`

. The rest of the column needs to use the map

The value of input for this map, the index, is in column `A`

. We want the value of the output for the map placed in column `B`

. In cell `B2`

, we want the output based on an input from `A2`

, so we type `=(3*A2)/(4*A2+5)`

.

We want to have this process repeated for the rest of the column, but we do not want to type a formula for each cell. We want the software to fill the column automatically. Notice that if you copy the formula from `B2`

and paste it into `B3`

, the formula is automatically adjusted to refer to `A3`

instead of `A2`

. This type of automatic modification of a formula is called relative addressing, where a pasted formula uses the relative position of a calculation. In this case, the relative position is to use the cell immediately to the left of the output.

Pasting the formula into every cell in the column is faster than typing a formula in every cell, but it is still too much work. We let the spreadsheet do all of the work by *filling* the remaining cells at once. We can do this by repeating the process earlier. Select the cell with a valid formula in `B2`

, and you will again see the grab box in the corner. Click on the box and drag down the column. When you release the selection, the formula will be filled into all of the selected cells, adjusted to use relative addresses.

You now have two columns: the index in column `A`

and the values in column `B`

. You can create a graph by selecting the two columns of data and inserting a scatter plot.