Understanding Geometric Sequences: Recognizing and Representing
Learn how to recognize and represent geometric sequences, understand the common ratio, explicit and recursive formulas, and solve for specific terms using formulas. Discover the steps to create and solve geometric sequences efficiently.
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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. E Exponential graphs looks like a J curve. X X P P O O N N E E N N T T I I A A L 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
Formulas and their Purpose Geometric Sequences Explicit Formula: ?? = ?? ?? ? Finds a specific term First Term Term Current Common Ratio Previous Term Recursive Formula: ?? = ?? ? r Uses previous terms to find the next terms 5 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 1stterm. 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, 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, 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 ?? = ?? ? ? 8 3.11: Geometric Sequences
Guided Practice Example 2 Create the recursive formula that defines the sequence: A geometric sequence is defined by 45, -15, 5, 5 3, 9 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, 5 3, Step 1: Find the common ratio Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? 10 3.11: Geometric Sequences
You Try 1 Use the following sequence to create a recursive formula. ??= ?? ? ? 10, -30, 90, -270, Step 1: Find the common ratio Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? 11 3.11: Geometric Sequences
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 sequence. Using the recursive formula: 1 10, with a1 = 3000. Find the first 5 terms of the a1 = 3000 a2 = a1 a2 = 3000 a3 = 300 a4 = 30 a5 = 3 1 10 1 10 = 300 1 10 = 30 1 10 = 3 1 10 = 3 10 The first five terms of the sequence are: 3000, 300, 30, 3, and 3 10. 13 3.11: Geometric Sequences
You Try 2 An arithmetic sequence is defined recursively by ?? = ?? 1 6, with a1 = 0.2 Find the first 5 terms of the sequence. 14 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. 15 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 & ??. Step 3: Substitute what you have. ?? = ? ?? ? ??? ???? ??? ????=?? ??? ?? ?? ??? ????? ???? ????? ?? ?. =? ?= ? ?? Step 2: Explicit or Recursive Formula? Step 4: Evaluate for specific term. We will use the explicit formula since we are finding a specific term. ??? = ? ??? ? = ? ?? = ???,??? So 524,288 is the 10th term in the sequence. ?? = ?? ?? ? 16 3.11: Geometric Sequences
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 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 1: Find the common ratio & ?? Step 3: Substitute what you have Step 2: Explicit or Recursive Formula? Step 4: Evaluate for specific term 18 3.11: Geometric Sequences
You Try 3 Use the following sequence to create an explicit formula.??= ?? ?? ? Then find ?14. - 4, 8, -16, 32, Step 1: Find the common ratio & ?? 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
