### Learning Outcomes

- Find the common ratio of a geometric sequence
- List terms of a geometric sequence given the first term and the common ratio

## 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.

### 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.

- 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: Finding Common Ratios

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

- [latex]1,2,4,8,16,\dots[/latex]
- [latex]48,12,4,2,\dots[/latex]

### Q & A

#### If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?

*No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.*

### Try It

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

[latex]5,10,15,20,\dots[/latex]

https://ohm.lumenlearning.com/multiembedq.php?id=68722&theme=oea&iframe_resize_id=mom2

### Try It

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

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

## 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,-8,-32,-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.

### tip for success

You can also use the definition of a geometric sequence, given at the beginning of the section, to write out several terms.

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

Try it on the example and practice problems below.

### Example: 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=\dfrac{1}{3}[/latex].

https://ohm.lumenlearning.com/multiembedq.php?id=156689&theme=oea&iframe_resize_id=mom3