Geometric Sequences

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.

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

[latex]\left\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\right\}[/latex].

How To: Given a set of numbers, determine if they represent a geometric sequence.

  1. Divide each term by the previous term.
  2. 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.

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

Try It

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

[latex]100,20,4,\frac{4}{5},\dots[/latex]

Try It

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.

[latex]\begin{align}&{a}_{1}=-2 \\ &{a}_{2}=\left(-2\cdot 4\right)=-8 \\ &{a}_{3}=\left(-8\cdot 4\right)=-32 \\ &{a}_{4}=\left(-32\cdot 4\right)-128\end{align}[/latex]

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.

  1. Multiply the initial term, [latex]{a}_{1}[/latex], by the common ratio to find the next term, [latex]{a}_{2}[/latex].
  2. 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.
  3. 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].

Try It

List the first five terms of the geometric sequence with [latex]{a}_{1}=18[/latex] and [latex]r=\frac{1}{3}[/latex].

Try It

Using Explicit Formulas for Geometric Sequences

We can write explicit formulas that allow us to find particular terms.

[latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex]

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

[latex]{a}_{n}=18\cdot {2}^{n - 1}[/latex]

A General Note: Explicit Formula for a Geometric Sequence

The nth term of a geometric sequence is given by the explicit formula:

[latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex]

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].

Try It

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]

Try It

Write an explicit formula for the following geometric sequence.

[latex]\left\{-1\text{, }3\text{, }-9\text{, }27\text{, }\dots\right\}[/latex]

Try It

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:

[latex]{a}_{n}={a}_{0}{r}^{n}[/latex]

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.

  1. Write a formula for the student population.
  2. Estimate the student population in 2020.

Try It

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.