Lesson 3.12
Concept: Geometric Sequences
EQ: How do we recognize and represent
geometric sequences? F.BF.1-2 & F.LE.2
Vocabulary: Geometric Sequence, Common
ratio, Explicit formula, Recursive formula
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3.11: Geometric Sequences
Activator: First Word
Using the word ‘EXPONENTIAL’, create a phrase starting with
each letter in the word on a sheet of paper. To get you started, I will
give you an example.
Exponential graphs looks like a ‘J’ curve.
X
P
O
N
E
N
T
I
A
L
Now you finish the rest.
2
3.8.2: Geometric Sequences
Introduction
• A geometric sequence is a list of terms
separated by a common ratio, r, which is the
number multiplied by each consecutive term in
a geometric sequence.
• A geometric sequence is an exponential
function with a domain of whole numbers in
which the ratio between any two consecutive
terms is equal.
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3.11: Geometric Sequences
Introduction (continued)
Just like arithmetic sequences, Geometric sequences
can be represented by formulas, either explicit or
recursive, and those formulas can be used to find a
certain term of the sequence or the number of a certain
value in the sequence.
Recall
• A recursive formula is a formula used to find the
next term of a sequence when the previous term is
known.
• An explicit formula is a formula used to find the nth
term of a sequence.
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3.11: Geometric Sequences
Formulas and their Purpose
Geometric Sequences
Explicit Formula: 𝑨𝒏 = 𝒂𝟏 · 𝒓𝒏−𝟏
“Finds a specific term”
First Term
Current
Term
Common Ratio
Previous
Term
Recursive Formula: 𝑨𝒏 = 𝑨𝒏−𝟏 · r
“Uses previous terms to find the next terms”
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3.11: Geometric Sequences
Steps to create formulas and solve for
geometric sequences
1. Find the common ratio by dividing the 2nd
term by the 1st term.
2. Decide which formula to use. (explicit or
recursive)
3. Substitute your values to create your
formula.
4. Find the specific term if asked to do so.
3.8.2: Geometric Sequences
6
Guided Practice
Example 1
Create the recursive formula that defines the
sequence:
A geometric sequence is defined by
2, 8, 32, 128, …
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3.11: Geometric Sequences
Guided Practice
Example 1, continued
Create the recursive formula that defines the
sequence:
A geometric sequence is defined by 2, 8, 32, 128, …
Step 1: Find the common ratio.
𝟐𝒏𝒅 𝒕𝒆𝒓𝒎 𝑨𝟐 𝟖
=
= =𝟒
𝟏𝒔𝒕 𝒕𝒆𝒓𝒎 𝑨𝟏 𝟐
Step 3: Substitute what you have.
Since r = 4 then
𝑨𝒏 = 𝑨𝒏−𝟏 ∙ 𝟒
Step 2: Explicit or Recursive Formula?
We will use the recursive formula which is
𝑨𝒏 = 𝑨𝒏−𝟏 ∙ 𝒓
3.11: Geometric Sequences
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Guided Practice
Example 2
Create the recursive formula that defines the
sequence:
A geometric sequence is defined by
45, -15, 5,
5
− ,
3
…
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3.11: Geometric Sequences
Guided Practice
Example 2, continued
Create the recursive formula that defines the
sequence:
5
3
A geometric sequence is defined by 45, -15, 5, − , …
Step 1: Find the common ratio
Step 3: Substitute what you have
Step 2: Explicit or Recursive Formula?
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3.11: Geometric Sequences
You Try 1
Use the following sequence to create a
recursive formula. 𝑨𝒏 = 𝑨𝒏−𝟏 · 𝒓
10, -30, 90, -270, …
Step 1: Find the common ratio
Step 3: Substitute what you have
Step 2: Explicit or Recursive Formula?
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3.11: Geometric Sequences
Guided Practice
Example 3
A geometric sequence is defined recursively by
an = an – 1 · −𝟑 , with a1 = 6. Find the first 5 terms of
the sequence.
Using the recursive formula:
a1 = 6
a2 = a1 · −𝟑
a2 =
a3 =
a4 =
a5 =
The first five terms of the sequence are:
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3.11: Geometric Sequences
Guided Practice – Example 4
A geometric sequence is defined recursively by
an = an – 1 ·
1
, with a1 = 3000. Find the first 5 terms of the
sequence.
Using the recursive formula:
a1 = 3000
10
a2 = a1 ·
1
10
1
10
1
·
10
1
·
10
1
a2 = 3000 ·
= 300
a3 = 300
= 30
a4 = 30
=3
3
a5 = 3 · =
10
10
The first five terms of the sequence are:
3000, 300, 30, 3, and
3.11: Geometric Sequences
3
.
10
13
You Try 2
An arithmetic sequence is defined recursively by
𝑎𝑛 = 𝑎𝑛−1 · 6, with a1 = 0.2
Find the first 5 terms of the sequence.
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3.11: Geometric Sequences
Guided Practice
Example 5
Write an explicit formula to represent the
sequence from example 1, and find the 10th
term.
The first five terms of the sequence are:
2, 8, 32, 128, and 512.
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3.11: Geometric Sequences
Guided Practice: Example 5, continued
The first five terms of the sequence are:
2, 8, 32, 128, and 512.
Step 1: Find the common ratio & 𝒂𝟏 .
𝟐𝒏𝒅 𝒕𝒆𝒓𝒎 𝒂𝟐 𝟖
=
= =𝟒
𝟏𝒔𝒕 𝒕𝒆𝒓𝒎
𝒂𝟏 𝟐
𝒂𝒏𝒅 𝒂𝟏 𝒊𝒔 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝒕𝒆𝒓𝒎 𝒘𝒉𝒊𝒄𝒉 𝒊𝒔 𝟐.
Step 2: Explicit or Recursive Formula?
We will use the explicit formula since we
are finding a specific term.
𝑨𝒏 = 𝒂𝟏 · 𝒓𝒏−𝟏
3.11: Geometric Sequences
Step 3: Substitute what you have.
𝑨𝒏 = 𝟐 · 𝟒𝒏−𝟏
Step 4: Evaluate for specific term.
𝑨𝟏𝟎 = 𝟐 · 𝟒𝟏𝟎−𝟏
= 𝟐 ⋅ 𝟒𝟗
= 𝟓𝟐𝟒, 𝟐𝟖𝟖
So 524,288 is the 10th term in the sequence.
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Guided Practice
Example 6
Write an explicit formula to represent the sequence from
example 3, and find the 15th term.
The first five terms of the sequence are:
6, -18, 54, -162, and 486
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3.11: Geometric Sequences
Guided Practice: Example 6, continued
The first five terms of the sequence are:
6, -18, 54, -162, and 486
Step 1: Find the common ratio & 𝒂𝟏
Step 3: Substitute what you have
Step 2: Explicit or Recursive Formula?
Step 4: Evaluate for specific term
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3.11: Geometric Sequences
You Try 3
Use the following sequence to create an explicit
formula. 𝒂𝒏 = 𝒂𝟏 · 𝒓𝒏−𝟏 Then find 𝑎14 .
- 4, 8, -16, 32, …
Step 1: Find the common ratio & 𝒂𝟏
Step 3: Substitute what you have
Step 2: Explicit or Recursive Formula?
Step 4: Evaluate for specific term
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3.11: Geometric Sequences
Summary: Last word
Using the word ‘GEOMETRIC’, create a
phrase with each letter just like with
exponential from before.
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3.8.2: Geometric Sequences