Definition of Geometric Sequences

The geometric sequence is another basic type of sequence. Understanding its definition is the first step toward mastering the laws of exponential growth.

Definition

In mathematical language: for a sequence $\{a_n\}$, if

$\frac{a_{n+1}}{a_n} = q \quad (n \in \mathbb{N}^*, q \neq 0 \text{ is a constant})$

then $\{a_n\}$ is a geometric sequence.

Methods of Determination

There are several methods to determine whether a sequence is a geometric sequence:

Method 1: The Definition Method

Check whether the ratio between adjacent terms is constant.

Example 1: Determine whether the sequence $2, 6, 18, 54, \ldots$ is a geometric sequence.

Solution:

• $\frac{a_2}{a_1} = \frac{6}{2} = 3$
• $\frac{a_3}{a_2} = \frac{18}{6} = 3$
• $\frac{a_4}{a_3} = \frac{54}{18} = 3$

The ratio between adjacent terms is always 3, so this is a geometric sequence with common ratio 3.

Method 2: The General Term Method

If the general term formula of a sequence can be written as $a_n = c \cdot q^n$ or $a_n = c \cdot q^{n-1}$ (where $c, q$ are nonzero constants), then the sequence is a geometric sequence.

Example 2: Is the sequence $a_n = 3 \cdot 2^{n-1}$ a geometric sequence?

Solution:

This is of the form $c \cdot q^{n-1}$ with $c = 3, q = 2$, so it is a geometric sequence with common ratio $q = 2$.

Verification: $a_1 = 3, a_2 = 6, a_3 = 12, \ldots$, indeed a geometric sequence with common ratio 2.

The Meaning of the Common Ratio

The common ratio $q$ determines the trend of the geometric sequence:

• $q > 1$: the sequence is increasing (e.g., $2, 4, 8, 16, \ldots$)
• $0 < q < 1$: the sequence is decreasing (e.g., $8, 4, 2, 1, \ldots$)
• $q < 0$: the sequence alternates in sign (e.g., $1, -2, 4, -8, \ldots$)
• $q = 1$: the sequence is constant (e.g., $5, 5, 5, 5, \ldots$)

Special Cases

A Sequence with Only One Term

A sequence with only one term can be regarded as a geometric sequence with any common ratio, because there are no “adjacent terms” to determine the common ratio.

A Sequence with Only Two Terms

Any two-term sequence (both nonzero) is a geometric sequence, with common ratio $q = \frac{a_2}{a_1}$.

A Sequence Containing Zero

If a sequence contains a zero term, then it is not a geometric sequence, because the ratio is meaningless.

Practice Problems

Exercise 1

Determine whether the following sequences are geometric sequences. If so, state the common ratio:

1. $3, 9, 27, 81, \ldots$
2. $1, 1, 1, 1, \ldots$
3. $1, 2, 3, 4, \ldots$
4. $16, 8, 4, 2, \ldots$

Exercise 2

Given that the general term of the sequence $\{a_n\}$ is $a_n = 5 \cdot (-2)^{n-1}$, determine whether the sequence is a geometric sequence. If so, find the common ratio.

Exercise 3

In the geometric sequence $\{a_n\}$, $a_2 = 6$ and $a_5 = 48$. Find the common ratio $q$.

Summary

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