Explainer: Geometric Sequences

In this explainer, we will learn how to calculate the common ratio, find next terms in a geometric sequence, and check if the sequence increases or decreases.

Definition: Geometric Sequences

A sequence is geometric if there is a common ratio between any two consecutive terms.

A common example of a geometric sequence is 2,4,8,16,.

Here, each successive term is double the previous one. In other words, the ration between any two consecutive terms is 2, or we could say that the term-to-term rule is “×2.

It is worth noting that geometric sequences do not always increase in value. They can be decreasing and indeed alternating. An example of a decreasing geometric sequence is 48,24,12,6,3,32,, and example of an alternating geometric sequence is 3,6,12,24,48,.

The decreasing sequence has a common ratio of 12 and the alternating sequence has a common ratio of 2.

Before we go any further, we need to introduce some notation. The first term of a geometric sequence is called 𝑎 and the common ratio between two consecutive terms is 𝑟. The second term is calculated by multiplying the first by 𝑟, so 𝑎𝑟, and the third term is the second multiplied by 𝑟, so 𝑎𝑟. We can continue this further as seen below:

In order to calculate the common ratio of a geometric sequence we can divide any term from the sequence by the term that immediately precedes it. For example, we could divide the third term by the second term or the second term by the first term. Let us look at the sequence 2,6,18,54,.

If we divide the second term by the first term we get 62=3.

Similarly, if we divide the fourth term by the third term we get 5418=3.

For completeness, if we divide the third term by the second tem we get 186=3.

Each of these divisions produces the same ratio which is 3. Therefore, we can assume, at least for the terms given, that the sequence is geometric with a common ratio of 3. Let us look at an example.

Example 1: Finding the Common Ratio of a Geometric Sequence

Find the common ratio of the geometric sequence 𝑎=1156,152,352,952,2752.

Answer

As we are told that we are looking at a geometric sequence, we do not need to check that the ratio is consistent between every two consecutive terms. Instead, we can choose any two consecutive terms. Here, we will consider the third and fourth term (𝑎 and 𝑎) and calculate 𝑎÷𝑎: 𝑟=952÷352=952×523=3.

Hence, the common ratio of our sequence is 3.

We could also be asked to find successive terms in a geometric sequence. This involves, first, finding the common ratio, and then using this to calculate additional terms by multiplying the preceding term by 𝑟. We will demonstrate this in another example.

Example 2: Finding Successive Terms of a Geometric Sequence

Find the next four terms in the geometric sequence 29,583,1169,.

Answer

First, we need to establish the common ratio of the geometric sequence, so we divide two consecutive terms. In this case, we will divide 𝑎 by 𝑎: 𝑟=583÷29=583×129=23.

In order to calculate the next term of a geometric sequence, we multiply the term before it by 𝑟. In this case, we multiply 1169 by 23: 1169×23=23227.

To find the fifth term we multiply the fourth term by 𝑟 and similarly for the sixth term we multiply the fifth by 𝑟 and continue this for any successive terms: 23227×23=46481,46481×23=928243,928243×23=1,856729.

The next four terms of the sequence are, therefore, 23227,46481,928243,1,856729.

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