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Lesson 3.12
Concept: Geometric SequencesEQ: How do we recognize and represent geometric sequences? F.BF.1-2 & F.LE.2
Vocabulary: Geometric Sequence, Common ratio, Explicit formula,
Recursive formula1
3.11: Geometric Sequences
Activator: First WordUsing 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.XPONENTIALNow 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 SequencesExplicit Formula:
“Finds a specific term”
Recursive Formula: r
“Uses previous terms to find the next terms”5
3.11: Geometric Sequences
Current Term
Previous Term
Common Ratio
First Term
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.6
3.8.2: Geometric Sequences
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, …
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3.11: Geometric Sequences
Step 1: Find the common ratio. Step 3: Substitute what you have.
Since r = 4 then
Step 2: Explicit or Recursive Formula?
We will use the recursive formula
which is
Guided Practice
Example 2
Create the recursive formula that defines the sequence:
A geometric sequence is defined by
45, -15, 5, , …
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3.11: Geometric Sequences
Guided Practice
Example 2, continued
Create the recursive formula that defines the sequence:
A geometric sequence is defined by 45, -15, 5, , …
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3.11: Geometric Sequences
Step 1: Find the common ratio Step 3: Substitute what you have
Step 2: Explicit or Recursive Formula?
You Try 1Use the following sequence to create a recursive formula.
10, -30, 90, -270, …
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3.11: Geometric Sequences
Step 1: Find the common ratio Step 3: Substitute what you have
Step 2: Explicit or Recursive Formula?
Guided Practice
Example 3
A geometric sequence is defined recursively by an = an – 1 · , with a1 = 6. Find the first 5 terms of the
sequence.Using the recursive formula:
a1 = 6
a2 = a1 ·
a2 =
a3 =
a4 =
a5 =
The first five terms of the sequence are: 12
3.11: Geometric Sequences
Guided Practice – Example 4
A geometric sequence is defined recursively by an = an – 1 · , with a1 = 3000. Find the first 5 terms of the
sequence.Using the recursive formula:
a1 = 3000
a2 = a1 ·
a2 = 3000 · = 300
a3 = 300 · = 30
a4 = 30 · = 3
a5 = 3 · =
The first five terms of the sequence are:
3000, 300, 30, 3, and .13
3.11: Geometric Sequences
You Try 2An arithmetic sequence is defined recursively by
· 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.
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3.11: Geometric Sequences
Step 1: Find the common ratio & . Step 3: Substitute what you have.
Step 2: Explicit or Recursive Formula?
We will use the explicit formula since we are finding a specific term.
Step 4: Evaluate for specific term.
So 524,288 is the 10th term in the sequence.
Guided Practice
Example 6Write 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
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3.11: Geometric Sequences
Step 1: Find the common ratio & Step 3: Substitute what you have
Step 2: Explicit or Recursive Formula? Step 4: Evaluate for specific term
You Try 3Use the following sequence to create an explicit formula. Then find .
- 4, 8, -16, 32, …
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3.11: Geometric Sequences
Step 1: Find the common ratio & Step 3: Substitute what you have
Step 2: Explicit or Recursive Formula?
Step 4: Evaluate for specific term
Summary: Last wordUsing the word ‘GEOMETRIC’, create a phrase with each letter just like with exponential from before.
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3.8.2: Geometric Sequences
