Explainer: Finding the 𝑛th Term of a Geometric Sequence

In this explainer, we will learn how to write explicit and recursive formulas for geometric sequences to find the value of the 𝑛th 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 𝑛th term. If the first term of a geometric sequence is π‘Ž and the common ratio is π‘Ÿ, then the sequence can be described as shown.

It is worth noting that the exponent of π‘Ÿ is always one less than the position, for example, the fourth term, π‘ŽοŠͺ, can be expressed as π‘Žπ‘ŸοŠ©. We can, therefore, establish that the 𝑛th term of a geometric sequence, π‘ŽοŠ, is π‘Ž=π‘Žπ‘Ÿ.

For example, for the geometric sequence 12,βˆ’2,4,βˆ’8,… we can see that the first term, π‘Ž, is 12. The common ratio, π‘Ÿ, is calculated by dividing any given term by the term that preceeds it, for example, π‘Ÿ=π‘ŽΓ·π‘ŽοŠ©οŠ¨, so π‘Ÿ=4Γ·(βˆ’2)=βˆ’2.

We can then write the 𝑛th term, π‘ŽοŠ by substituting the values of π‘Ž and π‘Ÿ into the formula: π‘Ž=ο€Ό12(βˆ’2).

Once we have calculated the formula for π‘ŽοŠ, we can use it to calculate any given term in the sequence. For example, the 10th term, π‘ŽοŠ§οŠ¦, is calculated by substituting 𝑛=10: π‘Ž=ο€Ό12(βˆ’2)=βˆ’256.

Let us look at an example.

Example 1: Finding the General Term for a Geometric Sequence

Find, the general term of the geometric sequence βˆ’76,βˆ’38,βˆ’19,βˆ’192,….

Answer

The general term of a sequence is another name for the 𝑛th term, π‘ŽοŠ. Remember that π‘Ž=π‘Žπ‘Ÿ, so we need to identify π‘Ž and calculate the common ratio π‘Ÿ. The first term of the sequence is π‘Ž, which is βˆ’76 and we can calculate the common ratio by dividing two consecutive terms. Here, we will divide π‘ŽοŠ¨ by π‘ŽοŠ§: π‘Ÿ=βˆ’38Γ·(βˆ’76)=12.

We can then substitute these values into the formula of the general term: π‘Ž=βˆ’76ο€Ό12.

As we mentioned earlier, another way to represent geometric sequences is by a term-to-term rule. For example, the geometric sequence 4,2,12,14,… has a common ratio of 12 and hence a term-to-term rule of β€œΓ—12.” 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 π‘ŽοŠ, we can define a recurrence relationship linking this with π‘ŽοŠοŠ°οŠ§. If we again consider the sequence 4,2,12,14,… we can see that π‘Ž=12π‘ŽοŠ¨οŠ§ and similarly π‘Ž=12π‘Ž.

We can generalize this for any two consecutive terms in the sequence: π‘Ž=12π‘Ž.

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 π‘Ž=12π‘ŽοŠοŠοŠ±οŠ§ 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 486,162,54,18,6,2,23.

Answer

In order to find a recursive formula, we need to identify the common ratio of the sequence. To do this, we will divide π‘ŽοŠ¨ by π‘ŽοŠ§: π‘Ÿ=162Γ·486=13.

We can now note that the term-to-term rule is β€œΓ—13,” for example, π‘Ž=13π‘ŽοŠ¨οŠ§. Hence, π‘Ž=13π‘ŽοŠοŠ°οŠ§οŠ or, alternatively, π‘Ž=13π‘ŽοŠοŠοŠ±οŠ§.

The generalization for the recurrence relation describing a geometric sequence is π‘Ž=π‘Ÿπ‘Ž, where π‘Ÿ is the common ratio.

Key Points

  • The general term, or 𝑛th term, of a geometric sequence is π‘Ž=π‘Žπ‘ŸοŠοŠοŠ±οŠ§, where π‘Ž is the first term and π‘Ÿ is the common ratio.
  • The general recursive formula for a geometric sequence is π‘Ž=π‘Ÿπ‘ŽοŠοŠ°οŠ§οŠ, where π‘Ÿ is the common ratio.

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