Learning Outcomes
By the end of this section, you will be able to:
- Find the common ratio for a geometric sequence.
- Give terms of a geometric sequence.
- Write the formula for a geometric sequence.
Finding Common Ratios
A geometric sequence changes by a constant factor. 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.
![A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3675/2018/09/27183608/CNX_Precalc_Figure_11_03_0012.jpg)
A General Note: Definition of a Geometric Sequence
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 [latex]{a}_{1}[/latex] is the initial term of a geometric sequence and [latex]r[/latex] is the common ratio, the sequence will be
How To: Given a set of numbers, determine if they represent a geometric sequence.
- Divide each term by the previous term.
- Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.
Example 1: Finding Common Ratios
Is the sequence geometric? If so, find the common ratio.
- [latex]1\text{,}2\text{,}4\text{,}8\text{,}16\text{,}..[/latex].
- [latex]48\text{,}12\text{,}4\text{, }2\text{,}..[/latex].
Try It
Is the sequence geometric? If so, find the common ratio.
[latex]5,10,15,20,\dots.[/latex]
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Is the sequence geometric? If so, find the common ratio.
[latex]100,20,4,\frac{4}{5},\dots[/latex]
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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].
How To: Given the first term and the common factor, find the first four terms of a geometric sequence.
- Multiply the initial term, [latex]{a}_{1}[/latex], by the common ratio to find the next term, [latex]{a}_{2}[/latex].
- Repeat the process, using [latex]{a}_{n}={a}_{2}[/latex] to find [latex]{a}_{3}[/latex] and then [latex]{a}_{3}[/latex] to find [latex]{a}_{4,}[/latex] until all four terms have been identified.
- Write the terms separated by commons within brackets.
Example 2: Writing the Terms of a Geometric Sequence
List the first four terms of the geometric sequence with [latex]{a}_{1}=5[/latex] and [latex]r=-2[/latex].
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List the first five terms of the geometric sequence with [latex]{a}_{1}=18[/latex] and [latex]r=\frac{1}{3}[/latex].
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Using Explicit Formulas for Geometric Sequences
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. An explicit formula for this sequence is
A General Note: Explicit Formula for a Geometric Sequence
The nth term of a geometric sequence is given by the explicit formula:
Example 4: Writing Terms of Geometric Sequences Using the Explicit Formula
Given a geometric sequence with [latex]{a}_{1}=3[/latex] and [latex]{a}_{4}=24[/latex], find [latex]{a}_{2}[/latex].
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Given a geometric sequence with [latex]{a}_{2}=4[/latex] and [latex]{a}_{3}=32[/latex] , find [latex]{a}_{6}[/latex].
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]
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Write an explicit formula for the following geometric sequence.
[latex]\left\{-1\text{, }3\text{, }-9\text{, }27\text{, }\dots\right\}[/latex]
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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:
Example 7: Solving Application Problems with Geometric Sequences
In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.
- Write a formula for the student population.
- Estimate the student population in 2020.
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A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.
a. Write a formula for the number of hits.
b. Estimate the number of hits in 5 weeks.