In this explainer, we will learn how to write explicit and recursive formulas for geometric sequences to find the value of the th term in a geometric sequence and how to find a term’s order given its value.
Geometric sequences can be described by a term-to-term rule and a position-to-term rule. A position-to-term rule, as the name suggests, allows us to calculate a specific term of the sequence, for example, the 10th term. The position-to-term rule is often referred to as the term. If the first term of a geometric sequence is and the common ratio is , then the sequence can be described as shown.
It is worth noting that the exponent of is always one less than the position, for example, the fourth term, , can be expressed as . We can, therefore, establish that the term of a geometric sequence, , is
For example, for the geometric sequence we can see that the first term, , is . The common ratio, , is calculated by dividing any given term by the term that preceeds it, for example, , so
We can then write the term, by substituting the values of and into the formula:
Once we have calculated the formula for , we can use it to calculate any given term in the sequence. For example, the 10th term, , is calculated by substituting :
Let us look at an example.
Example 1: Finding the General Term for a Geometric Sequence
Find, the general term of the geometric sequence .
The general term of a sequence is another name for the th term, . Remember that so we need to identify and calculate the common ratio . The first term of the sequence is , which is and we can calculate the common ratio by dividing two consecutive terms. Here, we will divide by :
We can then substitute these values into the formula of the general term:
As we mentioned earlier, another way to represent geometric sequences is by a term-to-term rule. For example, the geometric sequence has a common ratio of and hence a term-to-term rule of “.” To formalize this notion we need to introduce the idea of recurrence relations. A recurrence relation, for a geometric sequence, tells us the relationship between a term in the sequence and the term that immediately succeeds it. For example, if we consider a general term , we can define a recurrence relationship linking this with . If we again consider the sequence we can see that and similarly
We can generalize this for any two consecutive terms in the sequence:
This is called a recurrence relation and it can be used to work out successive terms in the sequence. It is worth noting here that we could have written our recurrence relation as and also that recurrence relations are often referred to as recursive formulae.
Let us look at an example.
Example 2: Finding a Recurrence Relation for a Geometric Sequence
Find a recursive formula for the sequence .
In order to find a recursive formula, we need to identify the common ratio of the sequence. To do this, we will divide by :
We can now note that the term-to-term rule is “,” for example, . Hence, or, alternatively, .
The generalization for the recurrence relation describing a geometric sequence is where is the common ratio.
- The general term, or term, of a geometric sequence is , where is the first term and is the common ratio.
- The general recursive formula for a geometric sequence is , where is the common ratio.