5-Minute Check 1: 5-Minute Check 1 A B C D The graph of y = 4 x is shown. State the y -intercept. Then use the graph to approximate value of 4 0.6 ? Determine whether the data in the table display exponential behavior.
Splash Screen: Splash Screen Homework: Page 547 WM – 3 problems 33 – 69 odd CW – 7 problems HW – 18 problems
Lesson Menu: Lesson Menu Five-Minute Check (over Lesson 9–7) Then/Now New Vocabulary Example 1: Identify Geometric Sequences Example 2: Find Terms of Geometric Sequences Key Concept: n th term of a Geometric Sequence Example 3: Find the n th Term of a Geometric Sequence Example 4: Real-World Example: Graph a Geometric Sequence
Then/Now: Then/Now You related arithmetic sequences to linear functions. (Lesson 3–5) Identify and generate geometric sequences. Relate geometric sequences to exponential functions.
Vocabulary: Vocabulary geometric sequence common ratio
Example 1: Example 1 Identify Geometric Sequences A. Determine whether each sequence is arithmetic , geometric , or neither . Explain. 0, 8, 16, 24, 32, ... 0 8 16 24 32 8 – 0 = 8 Answer: The common difference is 8. So the sequence is arithmetic. 16 – 8 = 8 24 – 16 = 8 32 – 24 = 8
Example 1: Example 1 Identify Geometric Sequences B. Determine whether each sequence is arithmetic , geometric , or neither . Explain. 64, 48, 36, 27, ... 64 48 36 27 Answer: The common ratio is , so the sequence is geometric. __ 3 4 __ 3 4 ___ 48 64 = __ 3 4 ___ 36 48 = __ 3 4 ___ 27 36 =
Example 2: Example 2 Find Terms of Geometric Sequences A. Find the next three terms in the geometric sequence. 1, –8, 64, –512, ... 1 –8 64 –512 The common ratio is –8. = –8 __ 1 –8 ___ 64 –8 = –8 = –8 ______ –512 64 Step 1 Find the common ratio.
Example 2: Example 2 Find Terms of Geometric Sequences Step 2 Multiply each term by the common ratio to find the next three terms. 262,144 × (–8) × (–8) × (–8) Answer: The next 3 terms in the sequence are 4096; –32,768; and 262,144. –32,768 4096 –512
Example 2: Example 2 Find Terms of Geometric Sequences B. Find the next three terms in the geometric sequence. 40, 20, 10, 5, .... 40 20 10 5 Step 1 Find the common ratio. = __ 1 2 ___ 40 20 = __ 1 2 ___ 10 20 = __ 1 2 ___ 5 10 The common ratio is . __ 1 2
Example 2: Example 2 Find Terms of Geometric Sequences Step 2 Multiply each term by the common ratio to find the next three terms. 5 __ 5 2 __ 5 4 __ 5 8 × __ 1 2 × __ 1 2 × __ 1 2 Answer: The next 3 terms in the sequence are , __ 5 2 __ 5 4 , and . __ 5 8
Concept: Concept
Example 3: Example 3 Find the n th Term of a Geometric Sequence A. Write an equation for the n th term of the geometric sequence 1, –2, 4, –8, ... . The first term of the sequence is 1. So, a 1 = 1. Now find the common ratio. 1 –2 4 –8 = –2 ___ –2 1 = –2 ___ 4 –2 = –2 ___ –8 4 a n = a 1 r n – 1 Formula for the n th term a n = 1 ( –2 ) n – 1 a 1 = 1 and r = –2 The common ration is –2. Answer: a n = 1(–2) n – 1
Example 3: Example 3 Find the n th Term of a Geometric Sequence B. Find the 12 th term of the sequence. 1, –2, 4, –8, ... . a n = a 1 r n – 1 Formula for the n th term a 12 = 1 ( –2 ) 12 – 1 For the n th term, n = 12. = 1(–2) 11 Simplify. = 1(–2048) (–2) 11 = –2048 = –2048 Multiply. Answer: The 12 th term of the sequence is –2048.
Example 4: Example 4 Graph a Geometric Sequence ART A 50-pound ice sculpture is melting at a rate in which 80% of its weight remains each hour. Draw a graph to represent how many pounds of the sculpture is left at each hour. Compared to each previous hour, 80% of the weight remains. So, r = 0.80. Therefore, the geometric sequence that models this situation is 50, 40, 32, 25.6, 20.48,… So after 1 hour, the sculpture weighs 40 pounds, 32 pounds after 2 hours, 25.6 pounds after 3 hours, and so forth. Use this information to draw a graph.
Example 4: Example 4 Graph a Geometric Sequence Answer:
Slide 17: Your Turn
Example 1: A B C Example 1 A. Determine whether the sequence is arithmetic , geometric , or neither . 1, 7, 49, 343, ...
Example 1: A B C Example 1 B. Determine whether the sequence is arithmetic , geometric , or neither . 1, 2, 4, 14, 54, ...
Example 2: A B C D Example 2 A. Find the next three terms in the geometric sequence. 1, –5, 25, –125, ....
Example 2: A B C D Example 2 B. Find the next three terms in the geometric sequence. 800, 200, 50, , .... __ 2 25
Example 3: A B C D Example 3 A. Write an equation for the n th term of the geometric sequence 3, –12, 48, –192, ....
Example 3: A B C D Example 3 B. Find the 7 th term of this sequence using the equation a n = 3(–4) n – 1 .
Example 4: A B C D Example 4 Soccer A soccer tournament begins with 32 teams in the first round. In each of the following rounds, on half of the teams are left to compete, until only one team remains. Draw a graph to represent how many teams are left to compete in each round.
End of the Lesson: End of the Lesson Homework: Page 547 WM – 3 problems 33 – 69 odd CW – 7 problems HW – 18 problems