Finding Common Ratios

The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.

Example 1: Finding Common Ratios

Is the sequence geometric? If so, find the common ratio.

1. [latex]1\text{,}2\text{,}4\text{,}8\text{,}16\text{,}..[/latex].
2. [latex]48\text{,}12\text{,}4\text{, }2\text{,}..[/latex].

Show Solution

Divide each term by the previous term to determine whether a common ratio exists.

1. [latex]\begin{align}&\frac{2}{1}=2&& \frac{4}{2}=2&& \frac{8}{4}=2&& \frac{16}{8}=2 \end{align}[/latex]
   The sequence is geometric because there is a common ratio. The common ratio is 2.
2. [latex]\begin{align}&\frac{12}{48}=\frac{1}{4}&& \frac{4}{12}=\frac{1}{3}&& \frac{2}{4}=\frac{1}{2} \end{align}[/latex]
   The sequence is not geometric because there is not a common ratio.

Analysis of the Solution

The graph of each sequence is shown in Figure 1. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.

Figure 1

Writing Terms of Geometric Sequences

Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is [latex]{a}_{1}=-2[/latex] and the common ratio is [latex]r=4[/latex], we can find subsequent terms by multiplying [latex]-2\cdot 4[/latex] to get [latex]-8[/latex] then multiplying the result [latex]-8\cdot 4[/latex] to get [latex]-32[/latex] and so on.

The first four terms are [latex]\left\{-2\text{, }-8\text{, }-32\text{, }-128\right\}[/latex].

Using Recursive Formulas for Geometric Sequences

A recursive formula allows us to find any term of a geometric sequence by using the previous term. Each term is the product of the common ratio and the previous term. For example, suppose the common ratio is 9. Then each term is nine times the previous term. As with any recursive formula, the initial term must be given.

Example 3: Using Recursive Formulas for Geometric Sequences

Write a recursive formula for the following geometric sequence.

[latex]\left\{6\text{, }9\text{, }13.5\text{, }20.25\text{, }\dots\right\}[/latex]

Show Solution

The first term is given as 6. The common ratio can be found by dividing the second term by the first term.

[latex]r=\frac{9}{6}=1.5[/latex]

Substitute the common ratio into the recursive formula for geometric sequences and define [latex]{a}_{1}[/latex].

[latex]\begin{align}&{a}_{n}=r{a}_{n - 1}\\ &{a}_{n}=1.5{a}_{n - 1}\text{ for }n\ge 2\\ &{a}_{1}=6\end{align}[/latex]

Analysis of the Solution

The sequence of data points follows an exponential pattern. The common ratio is also the base of an exponential function as shown in Figure 2.

Figure 2

Using Explicit Formulas for Geometric Sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.

Let’s take a look at the sequence [latex]\left\{18\text{, }36\text{, }72\text{, }144\text{, }288\text{, }...\right\}[/latex]. This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is

The graph of the sequence is shown in Figure 3.

Figure 3

Example 6: Writing an Explicit Formula for the nth Term of a Geometric Sequence

Write an explicit formula for the [latex]n\text{th}[/latex] term of the following geometric sequence.

[latex]\left\{2\text{, }10\text{, }50\text{, }250\text{, }\dots\right\}[/latex]

Show Solution

The first term is 2. The common ratio can be found by dividing the second term by the first term.

[latex]\frac{10}{2}=5[/latex]

The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.

[latex]\begin{align}{a}_{n}&={a}_{1}{r}^{\left(n - 1\right)}\\ {a}_{n}&=2\cdot {5}^{n - 1}\end{align}[/latex]

The graph of this sequence in Figure 4 shows an exponential pattern.

Figure 4

Solving Application Problems with Geometric Sequences

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of [latex]{a}_{0}[/latex] instead of [latex]{a}_{1}[/latex]. In these problems, we can alter the explicit formula slightly by using the following formula:

Key Equations

Key Concepts

• A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
• The constant ratio between two consecutive terms is called the common ratio.
• The common ratio can be found by dividing any term in the sequence by the previous term.
• The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.
• A recursive formula for a geometric sequence with common ratio [latex]r[/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[/latex] for [latex]n\ge 2[/latex] .
• As with any recursive formula, the initial term of the sequence must be given.
• An explicit formula for a geometric sequence with common ratio [latex]r[/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex].
• In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[/latex].

Glossary

common ratio

the ratio between any two consecutive terms in a geometric sequence

geometric sequence

a sequence in which the ratio of a term to a previous term is a constant