 # Concept: Geometric Sequences

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Concept: Geometric Sequences
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 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. 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. 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. 3.11: Geometric Sequences

Formulas and their Purpose Recursive Formula: 𝑨𝒏 = 𝑨 𝒏−𝟏 · r
Geometric Sequences Explicit Formula: 𝑨𝒏 = 𝒂 𝟏 ·𝒓 𝒏−𝟏 “Finds a specific term” Recursive Formula: 𝑨𝒏 = 𝑨 𝒏−𝟏 · r “Uses previous terms to find the next terms” First Term Current Term Common Ratio Previous Term 3.11: Geometric Sequences

Steps to create formulas and solve for geometric sequences
Find the common ratio by dividing the 2nd term by the 1st term. Decide which formula to use. (explicit or recursive) Substitute your values to create your formula. Find the specific term if asked to do so. 3.8.2: Geometric Sequences

A geometric sequence is defined by
Guided Practice Example 1 Create the recursive formula that defines the sequence: A geometric sequence is defined by 2, 8, 32, 128, … 3.11: Geometric Sequences

We will use the recursive formula which is 𝑨𝒏 = 𝑨 𝒏−𝟏 ∙𝒓
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 𝟏𝒔𝒕 𝒕𝒆𝒓𝒎 = 𝑨 𝟐 𝑨 𝟏 = 𝟖 𝟐 =𝟒 Step 3: Substitute what you have. Since r = 4 then 𝑨𝒏 = 𝑨 𝒏−𝟏 ∙𝟒 Step 2: Explicit or Recursive Formula? We will use the recursive formula which is 𝑨𝒏 = 𝑨 𝒏−𝟏 ∙𝒓 3.11: Geometric Sequences

A geometric sequence is defined by
Guided Practice Example 2 Create the recursive formula that defines the sequence: A geometric sequence is defined by 45, -15, 5, − 5 3 , … 3.11: Geometric Sequences

A geometric sequence is defined by 45, -15, 5, − 5 3 , …
Guided Practice Example 2, continued Create the recursive formula that defines the sequence: A geometric sequence is defined by 45, -15, 5, − 5 3 , … Step 1: Find the common ratio Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? 3.11: Geometric Sequences

You Try 1 Use the following sequence to create a recursive formula. 𝑨 𝒏 = 𝑨 𝒏−𝟏 ·𝒓 10, -30, 90, -270, … Step 1: Find the common ratio Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? 3.11: Geometric Sequences

The first five terms of the sequence are:
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: 3.11: Geometric Sequences

The first five terms of the sequence are:
Guided Practice – Example 4 A geometric sequence is defined recursively by an = an – 1 · , with a1 = Find the first 5 terms of the sequence. Using the recursive formula: a1 = 3000 a2 = a1 · 1 10 a2 = 3000 · = 300 a3 = 300 · = 30 a4 = 30 · = 3 a5 = 3 · = 3 10 The first five terms of the sequence are: 3000, 300, 30, 3, and 3.11: Geometric Sequences

Find the first 5 terms of the sequence.
You Try 2 An arithmetic sequence is defined recursively by 𝑎𝑛 = 𝑎 𝑛−1 · 6, with a1 = 0.2 Find the first 5 terms of the sequence. 3.11: Geometric Sequences

The first five terms of the sequence are:
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. 3.11: Geometric Sequences

The first five terms of the sequence are:
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 & 𝒂 𝟏 . 𝟐𝒏𝒅 𝒕𝒆𝒓𝒎 𝟏𝒔𝒕 𝒕𝒆𝒓𝒎 = 𝒂 𝟐 𝒂 𝟏 = 𝟖 𝟐 =𝟒 𝒂𝒏𝒅 𝒂 𝟏 𝒊𝒔 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝒕𝒆𝒓𝒎 𝒘𝒉𝒊𝒄𝒉 𝒊𝒔 𝟐. Step 3: Substitute what you have. 𝑨 𝒏 =𝟐 ·𝟒 𝒏−𝟏 Step 2: Explicit or Recursive Formula? We will use the explicit formula since we are finding a specific term. 𝑨 𝒏 = 𝒂 𝟏 ·𝒓 𝒏−𝟏 Step 4: Evaluate for specific term. 𝑨 𝟏𝟎 =𝟐 ·𝟒 𝟏𝟎−𝟏 =𝟐⋅ 𝟒 𝟗 =𝟓𝟐𝟒,𝟐𝟖𝟖 So 524,288 is the 10th term in the sequence. 3.11: Geometric Sequences

The first five terms of the sequence are:
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 3.11: Geometric Sequences

The first five terms of the sequence are:
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 & 𝒂 𝟏 Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? Step 4: Evaluate for specific term 3.11: Geometric Sequences

You Try 3 Use the following sequence to create an explicit formula. 𝒂 𝒏 = 𝒂 𝟏 · 𝒓 𝒏−𝟏 Then find 𝑎 14 . 4, 8, -16, 32, … Step 1: Find the common ratio & 𝒂 𝟏 Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? Step 4: Evaluate for specific term 3.11: Geometric Sequences

Summary: Last word Using the word ‘GEOMETRIC’, create a phrase with each letter just like with exponential from before. 3.8.2: Geometric Sequences