Lesson 3 12 Concept Geometric Sequences EQ How

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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 1 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. 3 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. 4 3. 11: Geometric Sequences

First Term Current Term Common Ratio Previous Term 5 3. 11: Geometric Sequences

Steps to create formulas and solve for geometric sequences 1. Find the common ratio by dividing the 2 nd term by the 1 st 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, … 7 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, … 8 3. 11: Geometric Sequences

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Step 1: Find the common ratio Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? 10 3. 11: Geometric Sequences

Step 1: Find the common ratio Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? 11 3. 11: Geometric Sequences

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Guided Practice Example 5 Write an explicit formula to represent the sequence from example 1, and find the 10 th term. The first five terms of the sequence are: 2, 8, 32, 128, and 512. 15 3. 11: Geometric Sequences

Guided Practice: Example 5, continued The first five terms of the sequence are: 2, 8, 32, 128, and 512. 16 3. 11: Geometric Sequences

Guided Practice Example 6 Write an explicit formula to represent the sequence from example 3, and find the 15 th term. The first five terms of the sequence are: 6, -18, 54, -162, and 486 17 3. 11: Geometric Sequences

Guided Practice: Example 6, continued The first five terms of the sequence are: 6, -18, 54, -162, and 486 Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? Step 4: Evaluate for specific term 18 3. 11: Geometric Sequences

Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? Step 4: Evaluate for specific term 19 3. 11: Geometric Sequences

Summary: Last word Using the word ‘GEOMETRIC’, create a phrase with each letter just like with exponential from before. 20 3. 8. 2: Geometric Sequences