Analyze Geometric Sequences and Series

Transcription

1 23 a4, 2A2A; P4A, P4B TEKS Analyze Geometric Sequences and Series Before You studied arithmetic sequences and series Now You will study geometric sequences and series Why? So you can solve problems about sports tournaments, as in Ex 58 Key Vocabulary geometric sequence common ratio geometric series In a geometric sequence, the ratio of any term to the previous term is constant This constant ratio is called the common ratio and is denoted by r E XAMPLE Identify geometric sequences Tell whether the sequence is geometric a 4, 0, 8, 28, 40, b 625, 25, 25, 5,, Solution To decide whether a sequence is geometric, find the ratios of consecutive terms a a 2 } 5 } a 3 } } 5 } a 4 } } 5 } a 5 } } 5 } 40 5 } 0 a 4 2 a2 0 5 a3 8 9 a b c The ratios are different, so the sequence is not geometric a 2 } a 5 25 } } 5 a 3 } a } 25 5 } 5 a 4 } a3 5 5 } 25 5 } 5 a 5 } a4 5 } 5 c Each ratio is } 5, so the sequence is geometric GUIDED PRACTICE for Example Tell whether the sequence is geometric Explain why or why not 8, 27, 9, 3,, 2, 2, 6, 24, 20, 3 24, 8, 26, 32, 264, KEY CONCEPT For Your Notebook Rule for a Geometric Sequence Algebra Example The nth term of a geometric sequence with first term a and common ratio r is given by: r n 2 The nth term of a geometric sequence with a first term of 3 and common ratio 2 is given by: 5 3(2) n 2 80 Chapter 2 Sequences and Series

2 E XAMPLE 2 Write a rule for the nth term Write a rule for the nth term of the sequence Then find a 7 a 4, 20, 00, 500, b 52, 276, 38, 29, Solution a The sequence is geometric with first term a 5 4 and common ratio r 5 } So, a rule for the nth term is: 4 AVOID ERRORS In the general rule for a geometric sequence, note that the exponent is n 2, not n r n 2 Write general rule 5 4(5) n 2 Substitute 4 for a and 5 for r The 7th term is a 7 5 4(5) ,500 b The sequence is geometric with first term a 5 52 and common ratio r 5} } So, a rule for the nth term is: 52 2 r n 2 Write general rule } 2 2 n 2 Substitute 52 for a and 2 } 2 for r The 7th term is a } } 8 E XAMPLE 3 Write a rule given a term and common ratio One term of a geometric sequence is a The common ratio is r 5 2 a Write a rule for the nth term b Graph the sequence Solution a Use the general rule to find the first term r n 2 Write general rule a 4 r 4 2 Substitute 4 for n 2 (2) 3 Substitute 2 for a 4 and 2 for r 5 Solve for a So, a rule for the nth term is: r n 2 Write general rule 5 5(2) n 2 Substitute 5 for a and 2 for r b Create a table of values for the sequence The graph of the first 6 terms of the sequence is shown Notice that the points lie on an exponential curve This is true for any geometric sequence with r > 0 n n at classzonecom 23 Analyze Geometric Sequences and Series 8

3 E XAMPLE 4 Write a rule given two terms Two terms of a geometric sequence are a and a Find a rule for the nth term Solution STEP Write a system of equations using r n 2 and substituting 3 for n (Equation ) and then 6 for n (Equation 2) a 3 r r 2 Equation a 6 r r 5 Equation 2 STEP 2 Solve the system 248 } r 2 Solve Equation for a } r 2 (r 5 ) Substitute for a in Equation r 3 Simplify 24 5 r Solve for r 248 (24) 2 Substitute for r in Equation 23 Solve for a STEP 3 Find a rule for r n 2 Write general rule 523(24) n 2 Substitute for a and r GUIDED PRACTICE for Examples 2, 3, and 4 Write a rule for the nth term of the geometric sequence Then find a 8 4 3, 5, 75, 375, 5 a , r a 2 522, a GEOMETRIC SERIES The expression formed by adding the terms of a geometric sequence is called a geometric series The sum of the first n terms of a geometric series is denoted by S n You can develop a rule for S n as follows S n a r a r 2 a r 3 a r n 2 2rS n 5 2 a r 2 a r 2 2 a r a r n 2 2 a r n S n ( 2 r) a r n So, S n ( 2 r) ( 2 r n ) If r Þ, you can divide each side of this equation by 2 r to obtain the following rule for S n KEY CONCEPT For Your Notebook The Sum of a Finite Geometric Series The sum of the first n terms of a geometric series with common ratio r Þ is: S n 2 rn } 2 r 2 82 Chapter 2 Sequences and Series

4 E XAMPLE 5 Find the sum of a geometric series Find the sum of the geometric series a 5 4(3) r Identify first term 4(3) i 2 Identify common ratio S 6 2 r6 } 2 r 2 Write rule for S } Substitute 4 for a and 3 for r 5 86,093,440 Simplify c The sum of the series is 86,093,440 E XAMPLE 6 Use a geometric sequence and series in real life MOVIE REVENUE In 990, the total box office revenue at US movie theaters was about $502 billion From 990 through 2003, the total box office revenue increased by about 59% per year a Write a rule for the total box office revenue (in billions of dollars) in terms of the year Let n 5 represent 990 b What was the total box office revenue at US movie theaters for the entire period ? Solution a Because the total box office revenue increased by the same percent each year, the total revenues from year to year form a geometric sequence Use a and r to write a rule for the sequence 5 502(059) n 2 Write a rule for b There are 4 years in the period , so find S 4 S 4 2 r4 } 2 r (059)4 } ø 05 c The total movie box office revenue for the period was about $05 billion GUIDED PRACTICE for Examples 5 and 6 7 Find the sum of the geometric series 8 6(22) i 2 8 MOVIE REVENUE Use the rule in part (a) of Example 6 to estimate the total box office revenue at US movie theaters in Analyze Geometric Sequences and Series 83

5 23 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p WS for Exs 9, 49, and 59 5 TAKS PRACTICE AND REASONING Exs 27, 54, 55, 59, 63, and 64 5 MULTIPLE REPRESENTATIONS Ex 6 VOCABULARY Copy and complete: The constant ratio of consecutive terms in a geometric sequence is called the? 2 WRITING How can you determine whether a sequence is geometric? EXAMPLE on p 80 for Exs 3 4 IDENTIFYING GEOMETRIC SEQUENCES Tell whether the sequence is geometric Explain why or why not 3, 4, 8, 6, 32, 4 4, 6, 64, 256, 024, 5 26, 36, 6,, } 6, 6 } 3, 2 } 3, 4 } 3, 8 } 3, 6 } 3, 7 },, } 3, 2, } 5, 8 2}, } 3, 2} 3, }, 2} 3, , 5, 25, 25, 0625, 0 23, 26, 2, 24, 248, 24, 2, 236, 08, 2324, 2 02, 06, 8, 54, 62, 3 25, 0, 20, 40, 80, 4 075, 5, 225, 3, 375, EXAMPLE 2 on p 8 for Exs 5 27 WRITING RULES Write a rule for the nth term of the geometric sequence Then find a 7 5, 24, 6, 264, 6 6, 8, 54, 62, 7 4, 24, 44, 864, 8 7, 235, 75, 2875, 9 2, } 3, } 9, } 27, , 2} 6, } 2, 2} 24, , 2,, 05, , 06, 22, 24, 23 22, 208, 2032, 2028, 24 7, 242, 252, 252, 25 5, 24, 392, 20976, 26 20, 80, 270, 405, 27 TAKS REASONING What is a rule for the nth term of the geometric sequence 5, 20, 80, 320,? A 5 5(2) n 2 B 5 5(4) n 2 C 5 5(24) n 2 D 5 5(22) n 2 EXAMPLE 3 on p 8 for Exs WRITING RULES Write a rule for the nth term of the geometric sequence Then graph the first six terms of the sequence 28 a 5 5, r a 522, r a 2 5 6, r a 2 5 5, r 5 } 2 32 a 5 5, r 5 } 8 33 a 4 522, r 52 } 4 34 a , r a 2 5 8, r a , r 5 5 ERROR ANALYSIS Describe and correct the error in writing the rule for the nth term of the geometric sequence for which a 5 3 and r r n 38 5 ra n 2 5 3(2) n 5 2(3) n 2 84 Chapter 2 Sequences and Series

6 EXAMPLE 4 on p 82 for Exs WRITING RULES Write a rule for the nth term of the geometric sequence that has the two given terms 39 a 5 3, a a 5, a a 52 } 4, a a 3 5 0, a a , a a , a a , a a } 4, a } 6 47 a 4 5 6, a } 8 EXAMPLE 5 on p 83 for Exs FINDING SUMS Find the sum of the geometric series (2) i } 4 2 i (4) i } 2 2 i i i } 2 2 i (24) i 54 TAKS REASONING What is the sum of the geometric series 9 2(3) i 2? A 9,680 B 9,68 C 9,682 D 9, TAKS REASONING Write a geometric series with 5 terms such that the sum of the series is 00 ( Hint: Choose a value of r and then find a ) 56 CHALLENGE Using the rule for the sum of a finite geometric series, write each polynomial as a rational expression a x x 2 x 3 x 4 b 3x 6x 3 2x 5 24x 7 PROBLEM SOLVING EXAMPLE 6 on p 83 for Exs SKYDIVING In a skydiving formation with R rings, each ring after the first has twice as many skydivers as the preceding ring The formation for R 5 2 is shown a Let be the number of skydivers in the nth ring Find a rule for b Find the total number of skydivers if there are R 5 4 rings Second ring First ring 58 SOCCER A regional soccer tournament has 64 participating teams In the first round of the tournament, 32 games are played In each successive round, the number of games played decreases by one half a Find a rule for the number of games played in the nth round For what values of n does your rule make sense? b Find the total number of games played in the regional soccer tournament 23 Analyze Geometric Sequences and Series 85

7 59 TAKS REASONING Abinary search technique used on a computer involves jumping to the middle of an ordered list of data (such as an alphabetical list of names) and deciding whether the item being searched for is there If not, the computer decides whether the item comes before or after the middle Half of the list is ignored on the next pass, and the computer jumps to the middle of the remaining list This is repeated until the item is found a Find a rule for the number of items remaining after the nth pass through an ordered list of 024 items b In the worst case, the item to be found is the only one left in the list after n passes through the list What is the worst-case value of n for a binary search of a list with 024 items? Explain 60 FRACTALS The Sierpinski carpet is a fractal created using squares The process involves removing smaller squares from larger squares First, divide a large square into nine congruent squares Remove the center square Repeat these steps for each smaller square, as shown below Assume that each side of the initial square is one unit long Stage Stage 2 Stage 3 a Let be the number of squares removed at the nth stage Find a rule for Then find the total number of squares removed through stage 8 b Let b n be the remaining area of the original square after the nth stage Find a rule for b n Then find the remaining area of the original square after stage 2 6 MULTIPLE REPRESENTATIONS Two companies, company A and company B, offer the same starting salary of $20,000 per year Company A gives a raise of $000 each year Company B gives a raise of 4% each year a Writing Rules Write rules giving the salaries and b n in the nth year at companies A and B, respectively Tell whether the sequence represented by each rule is arithmetic, geometric, or neither b Drawing Graphs Graph each sequence in the same coordinate plane c Finding Sums For each company, find the sum of wages earned during the first 20 years of employment d Using Technology Use a graphing calculator or spreadsheet to find after how many years the total amount earned at company B is greater than the total amount earned at company A 86 5 WORKED-OUT SOLUTIONS on p WS 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

8 62 CHALLENGE On January of each year, you deposit $2000 in an individual retirement account (IRA) that pays 5% annual interest You make a total of 30 deposits How much money do you have in your IRA immediately after you make your last deposit? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzonecom REVIEW Lesson 32; TAKS Workbook REVIEW Lesson 4; TAKS Workbook 63 TAKS PRACTICE The total cost of carnival tickets for 3 adults and 5 children is $49 The total cost of carnival tickets for 5 adults and 3 children is $55 What is the price, a, of one adult ticket and the price, c, of one child ticket? TAKS Obj 4 A a 5 $5; c 5 $8 B a 5 $725; c 5 $625 C a 5 $8; c 5 $5 D a 5 $0; c 5 $ TAKS PRACTICE What is the relationship between the graphs of y 5 3x 2 and y 5 5x 2? TAKS Obj 5 F G The graph of y 5 5x 2 is a reflection of the graph of y 5 3x 2 in the x-axis The graph of y 5 5x 2 is a 908 rotation of the graph of y 5 3x 2 about the origin H The graph of y 5 5x 2 is narrower than the graph of y 5 3x 2 J The graph of y 5 5x 2 is wider than the graph of y 5 3x 2 QUIZ for Lessons 2 23 Write the next term in the sequence Then write a rule for the nth term (p 794), 3, 5, 7, 2 25, 0, 25, 20, 3 }, 2 }, 3 }, 4 }, , 6, 64, 256, 5 2, 6, 2, 20, 6 9, 36, 8, 44, Find the sum of the series (p 794) 7 4 2i k 5 (k 2 3) 9 6 n 5 2 } n 2 Write a rule for the nth term of the arithmetic or geometric sequence Find a 5, then find the sum of the first 5 terms of the sequence 0, 7, 3, 9, (p 802) } 2, 2, 7 } 2, 5, (p 802) 2 5, 2, 2, 24, 27, (p 802) 3 2, 8, 32, 28, (p 80) 4 2, 4 } 3, 8 } 9, 6 } 27, (p 80) 5 23, 5, 275, 375, (p 80) 6 COLLEGE TUITION In 995, the average tuition at a public college in the United States was $2057 From 995 through 2002, the average tuition at public colleges increased by about 6% per year Write a rule for the average tuition in terms of the year Let n 5 represent 995 What was the average tuition at a public college in 2002? (p 80) EXTRA PRACTICE for Lesson 23, p 02 ONLINE QUIZ at classzonecom 87

9 MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons 2 23 MULTIPLE CHOICE TARGETS A target has red and blue rings that alternate in color The three innermost rings of the target are shown below Which expression is a series that gives the total area A of the target s n blue rings? TEKS a2 classzonecom 4 SEATING ARRANGEMENT At a restaurant, rectangular tables are placed together along their shared edges, as shown in the diagram below How many people can be seated around 8 tables arranged in this way? TEKS a ft The 3 innermost rings of the target A A 5 B A 5 C A 5 D A 5 n n n n (2i 2 )π (4i 2 3)π (4i 2 )π i 2 π 2 SALARY Maria has an annual salary of $45,000 during her first year of employment Her salary increases 35% per year What will Maria s salary be during her 5th year of employment? TEKS a F $49,672 G $5,24 H $5,639 J $53,446 3 CONSTRUCTION A staircase is being built that leads from the ground to an elevated deck The base of the staircase is a concrete slab that is 2 inches tall Each stair is 7 inches tall What is the height of the bottom of the 0th stair? TEKS a A 60 inches B 65 inches C 70 inches D 72 inches F 30 people G 32 people H 34 people J 36 people GRIDDED ANSWER STACKED PILES Pieces of chalk are stacked in a pile Part of the pile is shown below The bottom row has 5 pieces of chalk and the top row has 6 pieces of chalk Each row has one less piece of chalk than the row below it How many pieces of chalk are in the pile? TEKS a 6 RADIOACTIVE DECAY A scientist is studying the radioactive decay of Platinum-97 The scientist starts with a 66 gram sample of Platinum-97 and measures the amount remaining every two hours The recorded amounts (in grams) are 66, 33, 65, 825, After how many hours will the scientist first measure the sample and find that there is less than gram left? TEKS a 88 Chapter 2 Sequences and Series

10 Investigating g Algebra ACTIVITY Use before Lesson Investigating an Infinite Geometric Series TEKS MATERIALS scissors paper a4, a5; P4A QUESTION What is the sum of an infinite geometric series? You can illustrate an infinite geometric series by cutting a piece of paper into smaller and smaller pieces E XPLORE Model an infinite geometric series Start with a rectangular piece of paper Define its area to be square unit STEP Cut paper in half STEP 2 Cut paper again STEP 3 Repeat steps Fold the paper in half and cut along the fold Place one half on a desktop and hold the remaining half Fold the piece of paper you are holding in half and cut along the fold Place one half on the desktop and hold the remaining half Repeat Steps and 2 until you find it too difficult to fold and cut the piece of paper you are holding STEP 4 Find areas The first piece of paper on the desktop has an area of } square unit The second piece has an area of } square unit Write the areas 2 4 of the next three pieces of paper Explain why these areas form a geometric sequence STEP 5 Make a table Copy and complete the table by recording the number of pieces of paper on the desktop and the combined area of the pieces at each step Number of pieces Combined area } 2 }2 } 4 5??? DRAW CONCLUSIONS Use your observations to complete these exercises Based on your table, what number does the combined area of the pieces of paper appear to be approaching? 2 Using the formula for the sum of a finite geometric series, write and simplify a rule for the combined area A n of the pieces of paper after n cuts What happens to A n as n `? Justify your answer mathematically 24 Find Sums of Infinite Geometric Series 89