Geometric sequences

Geometric sequences are formed by a rule that lets us multiply each term by a fixed, non-zero number to find the next term. This constant factor is called the common ratio.

To identify a geometric sequence, check if you can get from one term to the next by multiplying the same constant factor every time. If you can, you’ve found a geometric sequence.

For example, in the sequence 2, 4, 8, 16, you’re multiplying by 2 each time, so it’s a geometric sequence.

Common Ratio

As we’ve said, the common ratio is the key to understanding geometric sequences. This is the value you multiply each term by to get the next term.

For example, in a sequence like 3, 6, 12, the common ratio is . Therefore, you multiply by 2 to get the next number in the sequence.

Let’s look at some more examples:

• In a sequence such as 4, 8, 16, the common ratio is . To find the next number in the sequence, you would multiply the last term by 2.

• For a sequence like 5, 10, 20, the common ratio is calculated as . To get the next term, you would multiply the last term by 2.

• In a sequence such as 7, 14, 28, the common ratio is . The next term can be found by multiplying the last term by 2.

Practice Questions

Question 1:

Find the common ratio of the sequence 5, 10, 20

Solution

The common ratio is

---

Question 2:

Is the sequence 4, 2, 1 a geometric sequence?

Solution

Yes, it is. The common ratio is