A geometric sequence is an ordered set, usually of numbers, such that 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 scientific or graphical display calculator, by typing in 1, enter, multiply (-3) enter, enter, enter…

# Fill in the Gaps

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

## Find the missing terms:

## First, calculate the ratio, then calculate the terms:

## Find a particular term:

# Formula

To find any term in a geometric sequence given the first term , and the common ratio , we can use the formula

.

Try this formula out with the applet ‘find a particular term’ above.

# 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.

In the following applet, the first term is 20. Try out:

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

Notice the various patterns made by the points.

What about , , ?

# Practice

Try the check your understanding questions on the CEMC courseware.

CK12 Practice test

Next: Geometric Series