Find Sums of Infinite Geometric Series

Transcription

1 a, AA; PB, PD TEKS Find Sums of Infinite Geometric Series Before You found the sums of finite geometric series Now You will find the sums of infinite geometric series Why? So you can analyze a fractal, as in Ex Key Vocabulary partial sum The sum S n of the first n terms of an infinite series is called a partial sum The partial sums of an infinite geometric series may approach a limiting value E XAMPLE Find partial sums onsider the infinite geometric series } } } 8 } 6 } Find and graph the partial sums S n for n,,,, and Then describe what happens to S n as n increases S } 0 S } } 07 S } } } < S } } } } ø S n S } } } 8 } 6 } ø 097 n From the graph, S n appears to approach as n increases at classzonecom SUMS OF INFINITE SERIES In Example, you can understand why S n approaches as n increases by considering the rule for S n : S n a } rn r } } n } } } n As n increases, } n approaches 0, so S n approaches Therefore, is defined to be the sum of the infinite geometric series in Example More generally, as n increases for any infinite geometric series with common ratio r between and, the value of S n a rn } r ø a 0 } r a } r 80 hapter Sequences and Series

2 KEY ONEPT For Your Notebook The Sum of an Infinite Geometric Series The sum of an infinite geometric series with first term a and common ratio r is given by S a } r provided r < If r, the series has no sum E XAMPLE Find sums of infinite geometric series Find the sum of the infinite geometric series a i (08) i b } 9 } 6 7 } 6 a For this series, a and r 08 b For this series, a and r } S a } r } 08 S a } r } } } 7 E XAMPLE TAKS PRATIE: Multiple hoice AVOID ERRORS If you substitute for a and for r in the formula S a }, you r get an answer of S } for the sum However, this answer is not correct because the sum formula does not apply when r What is the sum of the infinite geometric series 6 6? A } B } D Does not exist You know that a and a So, r } Because, the sum does not exist c The correct answer is D A B D GUIDED PRATIE for Examples,, and onsider the series } } 8 } 6 } 6 } Find and graph the partial sums S n for n,,,, and Then describe what happens to S n as n increases Find the sum of the infinite geometric series, if it exists n } n n } n } } 6 } 6 Find Sums of Infinite Geometric Series 8

3 E XAMPLE Use an infinite series as a model PENDULUMS A pendulum that is released to swing freely travels 8 inches on the first swing On each successive swing, the pendulum travels 80% of the distance of the previous swing What is the total distance the pendulum swings? The total distance traveled by the pendulum is: d 8 8(08) 8(08) 8(08) a } r Write formula for sum 8 } 08 Substitute 8 for a and 08 for r 90 Simplify c The pendulum travels a total distance of 90 inches, or 7 feet E XAMPLE Write a repeating decimal as a fraction Write 0 as a fraction in lowest terms 0 (00) (00) (00) a } r Write formula for sum (00) } 00 Substitute (00) for a and 00 for r 0 } 099 Simplify } 99 Write as a quotient of integers 8 } Reduce fraction to lowest terms c The repeating decimal 0 is 8 } as a fraction GUIDED PRATIE for Examples and WHAT IF? In Example, suppose the pendulum travels 0 inches on its first swing What is the total distance the pendulum swings? Write the repeating decimal as a fraction in lowest terms hapter Sequences and Series

4 EXERISES SKILL PRATIE HOMEWORK KEY WORKED-OUT SOLUTIONS on p WS for Exs, 7, and 9 TAKS PRATIE AND REASONING Exs,, 9, 0,,, and EXAMPLE on p 80 for Exs 6 VOABULARY opy and complete: The sum S n of the first n terms of an infinite series is called a(n)? WRITING Explain how to tell whether the series a r i has a sum i PARTIAL SUMS For the given series, find and graph the partial sums S n for n,,,, and Describe what happens to S n as n increases } } 6 } 8 } } 6 } } } 6 } } } 6 } 08 } } 6 6 } } } } } 6 EXAMPLES and on p 8 for Exs 7 FINDING SUMS Find the sum of the infinite geometric series, if it exists 7 n 8 } n 8 } i 6 i k 9() k 6 k n 6 } k 9 } n } i i 7 } } i i 0 k 7 8 } 9 k i 0 } 7 i 8 } k } 8 k } 0 } n n }6 () n n 0 9 ERROR ANALYSIS Describe and correct the error in finding the sum of the infinite geometric series n 7 } n For this series, a and r 7 } S a } r } 7 } } } } FINDING SUMS Find the sum of the infinite geometric series, if it exists 0 } 8 } } 8 } 7 } } 9 } 7 } 8 } } } 0 } 00 } } } EXAMPLE on p 8 for Exs REWRITING DEIMALS Write the repeating decimal as a fraction in lowest terms TAKS REASONING Which fraction is equal to the repeating decimal 8 88? A } B } 86 } 00 D } REASONING Show that 0999 is equal to Find Sums of Infinite Geometric Series 8

5 TAKS REASONING Find two infinite geometric series whose sums are each HALLENGE Specify the values of x for which the given infinite geometric series has a sum Then find the sum in terms of x x 6x 6x 6 6 } x } 8 x } x PROBLEM SOLVING EXAMPLE on p 8 for Exs TIRE SWING A person is given one push on a tire swing and then allowed to swing freely On the first swing, the person travels a distance of feet On each successive swing, the person travels 80% of the distance of the previous swing What is the total distance the person swings? 8 BUSINESS A company had a profit of $0,000 in its first year Since then, the company s profit has decreased by % per year If this trend continues, what is an upper limit on the total profit the company can make over the course of its lifetime? Justify your answer using an infinite geometric series 9 TAKS REASONING In 99, the number of cassette tapes shipped in the United States was million In each successive year, the number decreased by about 7% What is the total number of cassettes that will ship in 99 and after if this trend continues? A 0 million B 0 million 6 million D 9 billion 0 TAKS REASONING an the Greek hero Achilles, running at 0 feet per second, ever catch up to a tortoise that runs 0 feet per second if the tortoise has a 0 foot head start? The Greek mathematician Zeno said no He reasoned as follows: When Achilles runs 0 feet, Then, when Achilles gets to Achilles will keep halving the tortoise will be in a new that spot, the tortoise will the distance but will never spot, 0 feet away be feet away catch up to the tortoise In actuality, looking at the race as Zeno did, you can see that both the distances and the times Achilles required to traverse them form infinite geometric series Using the table, show that both series have finite sums Does Achilles catch up to the tortoise? Explain Distance (ft) Time (sec) WORKED-OUT SOLUTIONS on p WS TAKS PRATIE AND REASONING

6 TAKS REASONING A student drops a rubber ball from a height of 8 feet Each time the ball hits the ground, it bounces to 7% of its previous height a How far does the ball travel between the first and second bounces? between the second and third bounces? b Write an infinite series to model the total distance traveled by the ball, excluding the distance traveled before the first bounce c Find the total distance traveled by the ball, including the distance traveled before the first bounce d Show that if the ball is dropped from a height of h feet, then the total distance traveled by the ball (including the distance traveled before the first bounce) is 7h feet HALLENGE The Sierpinski triangle is a fractal created using equilateral triangles The process involves removing smaller triangles from larger triangles by joining the midpoints of the sides of the larger triangles as shown below Assume that the initial triangle has an area of square unit 8 ft 6 ft 6 ft ft ft 7 ft 7 ft ft ft Bounce number a Let a n be the total area of all the triangles that are removed at stage n Write a rule for a n b Find n problem? Stage Stage Stage a n What does your answer mean in the context of this MIXED REVIEW FOR TAKS TAKS PRATIE at classzonecom REVIEW Skills Review Handbook p 00; TAKS Workbook TAKS PRATIE Rectangle P represents 0 people who were surveyed about pet ownership ircle D represents the 7 people who said they owned a dog ircle represents the 0 people who said they owned a cat How many people do not own a dog or a cat? TAKS Obj 0 P D A 0 B 0 D 8 REVIEW Lesson ; TAKS Workbook TAKS PRATIE n PQR is a right triangle What is the length of } PR? TAKS Obj 6 F 0 cm H 0Ï } cm G 0Ï } cm J 0 cm R 0 cm P 608 P EXTRA PRATIE for Lesson, p 0 ONLINE QUIZ at classzonecom 8

7 Investigating g Algebra ATIVITY Exploring Recursive Rules MATERIALS computer with spreadsheet program Use before Lesson TEKS TEXAS classzonecom Keystrokes a, a, a6; PA QUESTION How can you evaluate a recursive rule for a sequence? A recursive rule for a sequence gives the beginning term or terms of the sequence and then an equation relating the nth term a n to one or more preceding terms For example, the rule a, a n a n 7 defines a sequence recursively EXPLORE Find terms of a sequence given by a recursive rule Find the first eight terms of the sequence defined by a, a n a n 7 What type of sequence does this rule represent? STEP Enter first term Enter the value of a into cell A STEP Enter recursive equation Enter the formula A7 into cell A STEP Fill cells Use the fill down command to copy the recursive equation into the rest of column A A A B A A7 A B A A77 A B STEP Identify terms and type of sequence The first eight terms of the sequence are,, 8,,, 9, 6, and This sequence is an arithmetic sequence because the difference of consecutive terms is always 7 DRAW ONLUSIONS Use your observations to complete these exercises Find the first eight terms of the sequence defined by a, a n 7a n What type of sequence does this rule represent? Write a recursive rule for the sequence,, 7,,,, Write a recursive rule for the sequence 8, 7, 9,,, }, What equation relates the nth term a n to the preceding term a n for an arithmetic sequence with common difference d? for a geometric sequence with common ratio r? 86 hapter Sequences and Series