Sequences

A mathematical sequence is a set of numbers which follows a pattern, for example 1, 2, 3, 4, 5, 6… (add 1 each time!), or 2, 4, 8, 16, 32…, and so on. A mathematical series is formed by adding all the terms in a sequence, for example 1/2 + 1/3 + 1/4 + 1/5 + 1/6…

Spreadsheets are very useful for generating and investigating sequences and series because they allow you to enter just a few starting numbers and then to generate the rest automatically. Excel uses the term 'series' for any set of numbers generated from a formula, regardless of whether they are summed.

Copy-Drag (or is it Drag-Copy?)

Before we generate a series we must learn a new technique for copying from one cell to another:

Enter any number (e.g. 7) in a cell and press the Enter key

Select the cell using the arrow keys or the mouse

Point at the small black marker in the bottom right corner of the cell and notice how the cursor changes to a black cross

Hold down the left mouse button and move the black cross down over the cells below the starting value

This copies the number into the cells you selected (you could also drag up, left or right with the same effect). This copying technique is very important and you should make sure you can do it smoothly.

Sequence 1..n

Here is a procedure to generate the simple sequence 1..n:

Enter 1 in the starting cell and 2 in the cell below

Select both cells with the mouse

Point at the small black marker in the bottom right corner of the selection and drag the black cross over a number of cells below.

You will see the next few numbers in the series appear in the cells below the 2. If you drag the mouse off the grid the worksheet will scroll until you push it back on to the grid. In this way you can create long series, should you need them.

Note that the series you generated is simple and additive. Try other starting values such as 1 and 3 and see what happens - you will get 1, 3, 5, 7… Excel notes the difference between the first two values and adds this to the current number to find the next one. A sequence formed by adding a constant value is called a Linear Series in Excel. If you plot the numbers on a line chart you will get a straight line.

Failed Geometric Sequences (Series)

Try the following:

Enter the numbers 1, 2, 4, 8, 16 in a column of 5 cells

Select all five numbers and use the copy marker as before to compute the next five values - drag it down over the next five cells.

You might have thought that you would get 32, 64, 128… but you don't. Excel computes the linear trend in the first five numbers and applies this to find the subsequent ones. This will prove useful later but it is not what we want now.

Successful Geometric Sequences (Series)

You can generate geometric or growth sequences (series) but not by dragging the copy marker:

Enter 1, 2, 4 in the first three cells of the series

Select the first three cells and the cells below where you want the next values to appear

Select Edit/Fill/Series, check the Growth option and enter 2 as the Step (effectively a multiplier).

Press Enter and the next numbers in the series will be filled in.

To produce the series 1, 3, 9, 27, 81… proceed as follows:

Enter 1, 3 in the first two cells of the series

Select these two cells and the cells below where you want the next values to appear

Select Edit/Fill/Series, check the Growth option and enter 3 as the Step

Press Enter and the next numbers in the series will be filled in.

A series such as 1, 2, 4, 8, 16… or 1, 3, 9, 27… in Excel is known as a Growth Series. If you plot them on a line chart you will get a curved line, not a straight one.

Other Series

Days, months. Not alphabetic.

Header and Footer

For practice add a header to the spreadsheet. Use a custom header and include your name, the centre number and today's date (automatic field).

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