Topics
5. Linear and Exponential Models
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United States of America
Grades 9-12

5.03 Geometric sequences

Lesson
Practice

Introduction

The last lesson focused on arithmetic sequences that have a recursive pattern of addition. In this lesson, we will discuss geometric sequences which have a recursive pattern of multiplication. We will explore how to identify geometric sequences and write them using recursive notation.

Geometric sequences

A geometric sequence is a recursive pattern of multiplication, where the same number is multiplied to each subsequent term.

Geometric sequence

A sequence of numbers in which each consecutive pair of numbers has a common ratio

Example:

1, \, 3, \, 9, \,27, \ldots

Geometric sequences all start with a first term, usually a_1, then either increase or decrease by a constant factor called the common ratio, denoted by r.

The size and sign of the common ratio play an important role in how the sequence grows. Geometric ratios greater than 1 will cause terms in the sequence to get larger. Ratios between 0 and 1 will cause the terms to get smaller. Negative ratios will cause sign changes across consecutive terms.

A geometric sequence is represented in recursive notation by the formula:

\displaystyle a_n=r\cdot a_{n-1}
\bm{a_n}
nth term
\bm{a_{n-1}}
previous term
\bm{r}
common ratio

Just like arithmetic sequences, the domain of any geometric sequence is a subset of the integers, usually starting from 0 or 1. If the domain begins from zero, we will be given a_0=c. If the sequence begins from one, we will be given a_1=c.

Examples

Example 1

Determine if the following sequences are geometric:

a

-3,\, 3,\, -3,\, 3,\, -3, \ldots

Worked Solution
Create a strategy

A sequence is geometric is the terms share a common ratio.

Apply the idea
\displaystyle \dfrac{3}{-3}\displaystyle =\displaystyle -1
\displaystyle \dfrac{-3}{3}\displaystyle =\displaystyle -1

This pattern continues, so all the terms do share a common ratio of -1. This sequence is geometric.

b
Three groups of connected squares are shown in the pattern. The group labeled Step 1 has 2 squares. The group labeled Step 2 has 4 squares. The group labeled Step 3 has 8 squares.
Worked Solution
Create a strategy

We need to look for a recursive pattern of multiplication between each step. For this one, we can count the squares in each step and determine if there is a common ratio.

Apply the idea

The first step has 2 squares. The second step has 4 squares. The third step has 8 squares.

The sequence for this pattern is 2,\,4,\,8,\ldots

Next, we need to look for a common ratio which we can do by dividing the terms.

4\div 2=2

8\div 4=2

The terms do share a common ratio, so this sequence is geometric.

c
1
2
3
4
5
n
20
25
30
35
40
45
50
55
60
a_n
Worked Solution
Create a strategy

Since geometric sequences increase or decrease by a common factor, they will form an exponential curve when graphed. Although these appear to form an exponential curve, we still need to check for a common ratio.

Apply the idea

The points on the curve are (1,60),\, (2,45),\, (3,35),\, (4,27),\, (5,21). We can check for a common ratio by dividing the outputs.

60\div 45=\dfrac{4}{3}

45\div35=\dfrac{9}{7}

35\div27=\dfrac{35}{27}

27\div 21=\dfrac{9}{7}

These terms do not share a common ratio, so the sequence is not geometric.

Example 2

A geometric sequence is defined by T_n=-2 \cdot T_{n-1} where T_1=\dfrac{1}{8}.

Find the next 4 terms of the sequence.

Worked Solution
Create a strategy

We already know the first term, so we need to find the 2nd through 5th terms. We can do that by using the rule with n=2,\,3,\,4,\,5.

Apply the idea

T_2=-2\times T_1=-2\times \dfrac{1}{8}=-\dfrac{1}{4}

T_3=-2\times T_2=-2\times -\dfrac{1}{4}=\dfrac{1}{2}

T_4=-2\times T_3=-2\times \dfrac{1}{2}=-1

T_5=-2\times T_4=-2\times -1=2

The next 4 terms of the sequence are -\dfrac{1}{4},\,\dfrac{1}{2},\,-1,\,2.

Reflect and check

If we didn't use the formula, we could have noticed from the formula that the common ratio is -2. That means we needed to multiply -2 from each term to get the next one.

Example 3

The first term of a geometric sequence is 5. The third term is 80.

a

Solve for the possible values of the common ratio, r, of this sequence.

Worked Solution
Create a strategy

In a geometric sequence, each term is the result of multiplying the previous term by a constant ratio. To get from the first term to the third term, we needed to multiply by the common ratio twice.

5\cdot r\cdot r=80

We can use this to set up an equation and solve for r.

Apply the idea
\displaystyle 5r^2\displaystyle =\displaystyle 80Given equation
\displaystyle r^2\displaystyle =\displaystyle 16Division property of equality
\displaystyle r\displaystyle =\displaystyle \pm 4Square root property

The common ratio is either 4 or -4.

Reflect and check

Either of these common ratios will satisfy the sequence with the information we know.

5\left(4\right)^2=5\left(16\right)=80

5\left(-4\right)^2=5\left(16\right)=80

To find one specific common ratio, we would need to know an even-numbered term.

b

State the recursive rule, T_n, that defines the sequence with a positive common ratio.

Worked Solution
Create a strategy

The positive common ratio was r=4, so we can substitute this into the recursive form of a geometric sequence.

Apply the idea

T_n=4\cdot T_{n-1} where T_1=5

c

State the recursive rule, T_n, that defines the sequence with a negative common ratio.

Worked Solution
Create a strategy

The negative common ratio was r=-4, so we can substitute this into the recursive form of a geometric sequence.

Apply the idea

T_n=-4\cdot T_{n-1} where T_1=5

Example 4

A group of students is working on a project that is due in 6 weeks. To determine how long it would take them to complete the project, they created little tasks that they could do along the way and came up with a total of 324 tasks. They worked on the project frequently at the beginning of the month but had to work on other projects later in the month.

The table below shows the number of tasks left to complete each week which can be represented by a geometric sequence.

n1234
a_n3241083612
a

Identify the common ratio.

Worked Solution
Create a strategy

We can find the common ratio by dividing a term by the previous term.

Apply the idea

Since we already know this is a geometric sequence, we can use any pair of values whose n-values are 1 unit apart. For example: \dfrac{12}{36}=\dfrac{1}{3}

The common ratio is \dfrac{1}{3}.

Reflect and check

We could have chosen other values and arrived at the same result. For example, \dfrac{108}{324}=\dfrac{1}{3} and \dfrac{36}{108}=\dfrac{1}{3}.

b

Determine the value of a_5.

Worked Solution
Create a strategy

Using the constant factor found in part (a), we know that as n increases by 1, a_n decreases by a factor of \dfrac{1}{3}. We can substitute r=\dfrac{1}{3} into the recursive formula and solve for the next term.

Apply the idea
\displaystyle a_n\displaystyle =\displaystyle \dfrac{1}{3} \cdot a_{n-1}Recursive formula
\displaystyle a_5\displaystyle =\displaystyle \dfrac{1}{3}\cdot a_4Substitute n=5
\displaystyle a_5\displaystyle =\displaystyle \dfrac{1}{3}\cdot 12Substitute a_4=12
\displaystyle a_5\displaystyle =\displaystyle 4Evaluate the multiplication

The 5th term is 4, so there are 4 tasks left to complete in the 5th week.

Reflect and check

You may have noticed that we are dividing each term by 3, but this is the same as multiplying each term by \dfrac{1}{3}.

324\div3=108

324\cdot \dfrac{1}{3}=108

Idea summary

The recursive formula for a geometric sequence is:

\displaystyle a_n=r\cdot a_{n-1}
\bm{a_n}
nth term
\bm{a_{n-1}}
previous term
\bm{r}
common ratio

Outcomes

F.BF.A.1

Write a function that describes a relationship between two quantities.

F.BF.A.1.A

Determine an explicit expression, a recursive process, or steps for calculation from a context.

F.BF.A.2

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

F.LE.A.1.C

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

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