Lesson 13.3: Geometric Sequences

New Concept

A geometric sequence is a list of numbers where each term is found by multiplying the previous one by a fixed number, the common ratio. We will explore how to find this ratio and use formulas to describe these sequences.

What’s next

Next, you'll master finding common ratios and terms through interactive examples. Then, you will apply recursive and explicit formulas in practice problems to solidify your understanding.

Property

A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If $a_1$ is the initial term of a geometric sequence and $r$ is the common ratio, the sequence will be

To determine if a set of numbers represents a geometric sequence, divide each term by the previous term. If the quotients are the same, a common ratio exists.

Property

The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. To find the first four terms given the first term and the common factor:

1. Multiply the initial term, $a_1$, by the common ratio to find the next term, $a_2$.
2. Repeat the process, using $a_n = a_{n-1} \cdot r$ to find $a_3$ and then $a_4$, until all four terms have been identified.
3. Write the terms separated by commas within brackets.

Examples

• List the first four terms of the geometric sequence with $a_1 = 3$ and $r = 4$. The terms are $\{3, 3 \cdot 4, 12 \cdot 4, 48 \cdot 4\}$, which gives $\{3, 12, 48, 192\}$.

• List the first four terms of the geometric sequence with $a_1 = 80$ and $r = \frac{1}{2}$. The terms are $\{80, 80 \cdot \frac{1}{2}, 40 \cdot \frac{1}{2}, 20 \cdot \frac{1}{2}\}$, which gives $\{80, 40, 20, 10\}$.

Property

The recursive formula for a geometric sequence with common ratio $r$ and first term $a_1$ is

To write this formula, you must state the initial term $a_1$, find the common ratio $r$ by dividing any term by its preceding term, and substitute $r$ into the formula.

Property

Because a geometric sequence is an exponential function whose domain is the set of positive integers, we can write explicit formulas that allow us to find particular terms. The $n$th term of a geometric sequence is given by the explicit formula:

Examples

• Write an explicit formula for the sequence $\{3, 12, 48, 192, \ldots\}$. The first term is $a_1=3$ and the common ratio is $r=\frac{12}{3}=4$. The formula is $a_n = 3 \cdot 4^{n-1}$.

Property

In real-world scenarios involving geometric sequences, we may need to use an initial term of $a_0$ instead of $a_1$. In these problems, we can alter the explicit formula slightly by using the following formula:

Examples

• A new car is purchased for 30,000 dollars and depreciates by 20% each year. Its value after $n$ years is modeled by $V_n = 30000 \cdot (1 - 0.20)^n = 30000 \cdot (0.80)^n$.