A geometric sequence is an ordered set, usually of numbers, where there is a common ratio between terms.

Here is an example:

Here is another example:

Notice that in the first example, the common ratio is 3. In the second example, the common ratio is -3.

You can create a sequence such as this on a graphical display calculator, by typing in 1, enter, multiply (-3) enter, enter, enter…

# 3 kinds of common ratio

- When the ratio is bigger than 1, the sequences gets bigger and bigger. We say it diverges.
- When the ratio is negative, the sequence oscillates between positive and negative values.
- When the ratio is between -1 and 1 (but not zero), the sequence diminishes. We say it converges to zero.

A geometric sequence can be defined using the first term and a common ratio. Type in your own value for first term in the box below; and your own common ratio. Slide the value ‘n’ to generate the first 20 terms and to draw a graph of your sequence on the axes. You will probably need to use the ‘move’ tool (four arrows) in the tool bar to adjust the y axis. Click on the ‘move’ tool, and put it over the y axis. Click and drag to adjust.

Here are some to try out:

- First term = 2; common ratio = 1.2 (ratio is positive, bigger than 1).
- First term = 2; common ratio = -1.2 (ratio is negative)
- First term = 2; common ratio = 0.5 (ratio is positive, less than 1)

Notice the patterns made by the points.

Did you try out a ratio value equal to zero or equal to one? What happens?

# Fill in the Gaps

The following applets generate geometric sequences. Using your own technique figure out the numbers in the gaps.

## Easiest

## A little more tricky

## Trickiest

I’m done! Take me back to the Sequences and Series Menu!