. Geometric Sequeces Essetial Questio How ca you use a geometric sequece to describe a patter? I a geometric sequece, the ratio betwee each pair of cosecutive terms is the same. This ratio is called the commo ratio. Describig Calculator Patters Work with a parter. Eter the keystrokes o a calculator ad record the results i the table. Describe the patter. a. Step 1 2 = b. Step 1 4 = Step 2 2 = Step 2. 5 = Step 3 2 = Step 3 Step 4 2 = Step 4 Step 5 2 = Step 5. 5 =. 5 =. 5 = Step 1 2 3 4 5 Calculator display c. Use a calculator to make your ow sequece. Start with ay umber ad multiply by 3 each time. Record your results i the table. Step 1 2 3 4 5 Calculator display Step 1 2 3 4 5 Calculator display LOOKING FOR REGULARITY IN REPEATED REASONING To be proficiet i math, you eed to otice whe calculatios are repeated ad look both for geeral methods ad for shortcuts. d. Part (a) ivolves a geometric sequece with a commo ratio of 2. What is the commo ratio i part (b)? part (c)? Foldig a Sheet of Paper Work with a parter. A sheet of paper is about 0.1 millimeter thick. a. How thick will it be whe you fold it i half oce? twice? three times? b. What is the greatest umber of times you ca fold a piece of paper i half? How thick is the result? c. Do you agree with the statemet below? Explai your reasoig. If it were possible to fold the paper i half 15 times, it would be taller tha you. Commuicate Your Aswer 3. How ca you use a geometric sequece to describe a patter? 4. Give a example of a geometric sequece from real life other tha paper foldig. Sectio. Geometric Sequeces 331
. Lesso What You Will Lear Core Vocabulary geometric sequece, p. 332 commo ratio, p. 332 Previous arithmetic sequece commo differece Idetify geometric sequeces. Exted ad graph geometric sequeces. Write geometric sequeces as fuctios. Idetifyig Geometric Sequeces Core Cocept Geometric Sequece I a geometric sequece, the ratio betwee each pair of cosecutive terms is the same. This ratio is called the commo ratio. Each term is foud by multiplyig the previous term by the commo ratio. 1, 5, 25, 125,... Terms of a geometric sequece 5 5 5 commo ratio Idetifyig Geometric Sequeces Decide whether each sequece is arithmetic, geometric, or either. Explai your reasoig. a. 120, 0, 30, 15,... b. 2,, 11, 17,... a. Fid the ratio betwee each pair of cosecutive terms. 120 0 30 15 0 120 = 1 30 2 0 = 1 15 2 30 = 1 2 The ratios are the same. The commo ratio is 1 2. So, the sequece is geometric. b. Fid the ratio betwee each pair of cosecutive terms. 2 11 17 2 = 3 11 = 1 5 17 11 = 1 11 There is o commo ratio, so the sequece is ot geometric. Fid the differece betwee each pair of cosecutive terms. 2 11 17 2 = 4 11 = 5 17 11 = There is o commo differece, so the sequece is ot arithmetic. So, the sequece is either geometric or arithmetic. Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com Decide whether the sequece is arithmetic, geometric, or either. Explai your reasoig. 1. 5, 1, 3, 7,... 2. 1024, 128, 1, 2,... 3. 2,, 10, 1,... 332 Chapter Expoetial Fuctios ad Sequeces
Extedig ad Graphig Geometric Sequeces Extedig Geometric Sequeces Write the ext three terms of each geometric sequece. a. 3,, 12, 24,... b. 4, 1, 4, 1,... Use tables to orgaize the terms ad exted each sequece. a. Positio 1 2 3 4 5 7 Term 3 12 24 48 9 192 Each term is twice the previous term. So, the commo ratio is 2. 2 2 2 2 2 2 Multiply a term by 2 to fid the ext term. The ext three terms are 48, 9, ad 192. LOOKING FOR STRUCTURE Whe the terms of a geometric sequece alterate betwee positive ad egative terms, or vice versa, the commo ratio is egative. b. Positio 1 2 3 4 5 7 Term 4 1 4 1 1 4 The ext three terms are 1 4, 1 1, ad 1 4. 1 1 ( 1 4) Graphig a Geometric Sequece 1 4 Multiply a term by 1 4 to fid the ext term. Graph the geometric sequece 32, 1, 8, 4, 2,.... What do you otice? STUDY TIP The poits of ay geometric sequece with a positive commo ratio lie o a expoetial curve. Make a table. The plot the ordered pairs (, ). Positio, 1 2 3 4 5 Term, 32 1 8 4 2 The poits appear to lie o a expoetial curve. 32 24 1 (1, 32) (2, 1) (3, 8) 8 (4, 4) (5, 2) 0 0 2 4 Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com Write the ext three terms of the geometric sequece. The graph the sequece. 4. 1, 3, 9, 27,... 5. 2500, 500, 100, 20,.... 80, 40, 20, 10,... 7. 2, 4, 8, 1,... Sectio. Geometric Sequeces 333
Writig Geometric Sequeces as Fuctios Because cosecutive terms of a geometric sequece have a commo ratio, you ca use the first term a 1 ad the commo ratio r to write a expoetial fuctio that describes a geometric sequece. Let a 1 = 1 ad r = 5. Positio, Term, Writte usig a 1 ad r Numbers 1 first term, a 1 a 1 1 2 secod term, a 2 a 1 r 1 5 = 5 3 third term, a 3 a 1 r 2 1 5 2 = 25 4 fourth term, a 4 a 1 r 3 1 5 3 = 125 th term, a 1 r 1 1 5 1 STUDY TIP Notice that the equatio = a 1 r 1 is of the form y = ab x. Core Cocept Equatio for a Geometric Sequece Let be the th term of a geometric sequece with first term a 1 ad commo ratio r. The th term is give by = a 1 r 1. Fidig the th Term of a Geometric Sequece Write a equatio for the th term of the geometric sequece 2, 12, 72, 432,.... The fid a 10. The first term is 2, ad the commo ratio is. = a 1 r 1 Equatio for a geometric sequece = 2() 1 Substitute 2 for a 1 ad for r. Use the equatio to fid the 10th term. = 2() 1 Write the equatio. a 10 = 2() 10 1 Substitute 10 for. = 20,155,392 Simplify. The 10th term of the geometric sequece is 20,155,392. Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com Write a equatio for the th term of the geometric sequece. The fid a 7. 8. 1, 5, 25, 125,... 9. 13, 2, 52, 104,... 10. 432, 72, 12, 2,... 11. 4, 10, 25, 2.5,... 334 Chapter Expoetial Fuctios ad Sequeces
You ca rewrite the equatio for a geometric sequece with first term a 1 ad commo ratio r i fuctio otatio by replacig with f (). f () = a 1 r 1 The domai of the fuctio is the set of positive itegers. Modelig with Mathematics Clickig the zoom-out butto o a mappig website doubles the side legth of the square map. After how may clicks o the zoom-out butto is the side legth of the map miles? Zoom-out clicks 1 2 3 Map side legth (miles) 5 10 20 USING APPROPRIATE TOOLS STRATEGICALLY You ca also use the table feature of a graphig calculator to fid the value of for which f () =. 3 4 5 7 X 9 X=8 Y1 20 40 80 10 320 1280 Y2 1. Uderstad the Problem You kow that the side legth of the square map doubles after each click o the zoom-out butto. So, the side legths of the map represet the terms of a geometric sequece. You eed to fid the umber of clicks it takes for the side legth of the map to be miles. 2. Make a Pla Begi by writig a fuctio f for the th term of the geometric sequece. The fid the value of for which f () =. 3. Solve the Problem The first term is 5, ad the commo ratio is 2. f () = a 1 r 1 Fuctio for a geometric sequece f () = 5(2) 1 Substitute 5 for a 1 ad 2 for r. The fuctio f () = 5(2) 1 represets the geometric sequece. Use this fuctio to fid the value of for which f () =. So, use the equatio = 5(2) 1 to write a system of equatios. y = 5(2) 1 Equatio 1 y = Equatio 2 The use a graphig calculator to graph the equatios ad fid the poit of itersectio. The poit of itersectio is (8, ). 1000 y = y = 5(2) 1 Itersectio 0 X=8 Y= 12 0 So, after eight clicks, the side legth of the map is miles. 4. Look Back Fid the value of for which f () = algebraically. = 5(2) 1 Write the equatio. 128 = (2) 1 Divide each side by 5. 2 7 = (2) 1 Rewrite 128 as 2 7. 7 = 1 Equate the expoets. 8 = Add 1 to each side. Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com 12. WHAT IF? After how may clicks o the zoom-out butto is the side legth of the map 250 miles? Sectio. Geometric Sequeces 335
. Exercises Dyamic Solutios available at BigIdeasMath.com Vocabulary ad Core Cocept Check 1. WRITING Compare the two sequeces. 2, 4,, 8, 10,... 2, 4, 8, 1, 32,... 2. CRITICAL THINKING Why do the poits of a geometric sequece lie o a expoetial curve oly whe the commo ratio is positive? Moitorig Progress ad Modelig with Mathematics I Exercises 3 8, fid the commo ratio of the geometric sequece. 3. 4, 12, 3, 108,... 4. 3,, 1, 1,... 5. 3, 3, 24, 192,.... 0.1, 1, 10, 100,... 8 7. 128, 9, 72, 54,... 8. 12, 54, 18,,... I Exercises 9 14, determie whether the sequece is arithmetic, geometric, or either. Explai your reasoig. (See Example 1.) 9. 8, 0, 8, 1,... 10. 1, 4, 7, 10,... 11. 9, 14, 20, 27,... 12. 3, 3 49 7, 3, 21,... 13. 192, 24, 3, 3,... 14. 25, 18, 11, 4,... 8 I Exercises 15 18, determie whether the graph represets a arithmetic sequece, a geometric sequece, or either. Explai your reasoig. 15. 17. 240 10 (4, 250) 1. 80 (1, 2) (3, 50) (2, 10) 0 0 2 4 120 80 40 (1, 120) 18. (2, 0) 0 0 2 4 (3, 20) (4, 5) 18 12 (4, 19) (3, 1) (1, 4) (2, 11) 0 0 2 4 24 12 0 12 (4, 24) (2, ) (3, 15) 2 4 (1, 3) I Exercises 19 24, write the ext three terms of the geometric sequece. The graph the sequece. (See Examples 2 ad 3.) 19. 5, 20, 80, 320,... 20. 3, 12, 48, 192,... 21. 81, 27, 9, 3,... 22. 375, 75, 15, 3,... 23. 32, 8, 2, 1 2,... 24. 1, 8 9, 4,,... 3 I Exercises 25 32, write a equatio for the th term of the geometric sequece. The fid a. (See Example 4.) 25. 2, 8, 32, 128,... 2. 0., 3, 15, 75,... 1 27. 8, 1 4, 1 2, 1,... 28. 0.1, 0.9, 8.1, 72.9,... 29. 30. 1 2 3 4 7 74 7.4 7.4 1 2 3 4 192 48 12 3 31. 32. (1, 0.5) (3, 18) 240 0 50 100 1 3 5 (2, 3) (4, 108) 10 33. PROBLEM SOLVING A badmito touramet begis with 128 teams. After the first roud, 4 teams remai. After the secod roud, 32 teams remai. How may teams remai after the third, fourth, ad fifth rouds? 80 (1, 224) (3, 5) (2, 112) 0 0 2 4 (4, 28) 33 Chapter Expoetial Fuctios ad Sequeces
34. PROBLEM SOLVING The graphig calculator scree 38. MODELING WITH MATHEMATICS You start a chai displays a area of 9 square uits. After you zoom out oce, the area is 384 square uits. After you zoom out a secod time, the area is 153 square uits. What is the scree area after you zoom out four times? email ad sed it to six frieds. The ext day, each of your frieds forwards the email to six people. The process cotiues for a few days. a. Write a fuctio that represets the umber of people who have received the email after days. b. After how may days will 129 people have received the email? 2 35. ERROR ANALYSIS Describe ad correct the error i writig the ext three terms of the geometric sequece. 8, 4, MATHEMATICAL CONNECTIONS I Exercises 39 ad 2, ( 2) ( 2) 1,... 40, (a) write a fuctio that represets the sequece of figures ad (b) describe the 10th figure i the sequece. ( 2) 39. The ext three terms are 2, 4, ad 8. 3. ERROR ANALYSIS Describe ad correct the error i writig a equatio for the th term of the geometric sequece. 40. 2, 12, 72, 432,... The first term is 2, ad the commo ratio is. a = a1r 1 a = 2( ) 1 41. REASONING Write a sequece that represets the 37. MODELING WITH MATHEMATICS The distace (i millimeters) traveled by a swigig pedulum decreases after each swig, as show i the table. (See Example 5.) Swig Distace (i millimeters) 1 2 3 25 500 400 umber of teams that have bee elimiated after rouds of the badmito touramet i Exercise 33. Determie whether the sequece is arithmetic, geometric, or either. Explai your reasoig. 42. REASONING Write a sequece that represets the perimeter of the graphig calculator scree i Exercise 34 after you zoom out times. Determie whether the sequece is arithmetic, geometric, or either. Explai your reasoig. 43. WRITING Compare the graphs of arithmetic sequeces to the graphs of geometric sequeces. 44. MAKING AN ARGUMENT You are give two cosecutive terms of a sequece. distace a. Write a fuctio that represets the distace the pedulum swigs o its th swig. b. After how may swigs is the distace 25 millimeters?..., 8, 0,... Your fried says that the sequece is ot geometric. A classmate says that is impossible to kow give oly two terms. Who is correct? Explai. Sectio. hsb_alg1_pe_00.idd 337 Geometric Sequeces 337 2/5/15 7:52 AM
45. CRITICAL THINKING Is the sequece show a arithmetic sequece? a geometric sequece? Explai your reasoig. 3, 3, 3, 3,... 4. HOW DO YOU SEE IT? Without performig ay calculatios, match each equatio with its graph. Explai your reasoig. a. = 20 ( 4 3 ) 1 b. = 20 ( 3 4 ) 1 51. REPEATED REASONING A soup kitche makes 1 gallos of soup. Each day, a quarter of the soup is served ad the rest is saved for the ext day. a. Write the first five terms of the sequece of the umber of fluid ouces of soup left each day. b. Write a equatio that represets the th term of the sequece. c. Whe is all the soup goe? Explai. A. 1 B. 200 52. THOUGHT PROVOKING Fid the sum of the terms of the geometric sequece. 8 100 1, 1 2, 1 4, 1 8,..., 1 2 1,... 0 0 4 8 0 0 4 8 Explai your reasoig. Write a differet ifiite geometric sequece that has the same sum. 47. REASONING What is the 9th term of the geometric sequece where a 3 = 81 ad r = 3? 48. OPEN-ENDED Write a sequece that has a patter but is ot arithmetic or geometric. Describe the patter. 49. ATTENDING TO PRECISION Are the terms of a geometric sequece idepedet or depedet? Explai your reasoig. 50. DRAWING CONCLUSIONS A college studet makes a deal with her parets to live at home istead of livig o campus. She will pay her parets $0.01 for the first day of the moth, $0.02 for the secod day, $0.04 for the third day, ad so o. a. Write a equatio that represets the th term of the geometric sequece. b. What will she pay o the 25th day? c. Did the studet make a good choice or should she have chose to live o campus? Explai. 53. OPEN-ENDED Write a geometric sequece i which a 2 < a 1 < a 3. 54. NUMBER SENSE Write a equatio that represets the th term of each geometric sequece show. 1 2 3 4 2 18 54 1 2 3 4 b 1 5 25 125 a. Do the terms a 1 b 1, a 2 b 2, a 3 b 3,... form a geometric sequece? If so, how does the commo ratio relate to the commo ratios of the sequeces above? a 2 a 3 b. Do the terms a 1,,,... form a geometric b 1 b 2 b 3 sequece? If so, how does the commo ratio relate to the commo ratios of the sequeces above? Maitaiig Mathematical Proficiecy Use residuals to determie whether the model is a good fit for the data i the table. Explai. (Sectio 4.5) 55. y = 3x 8 5. y = 5x + 1 Reviewig what you leared i previous grades ad lessos x 0 1 2 3 4 5 y 10 2 1 2 1 7 10 x 3 2 1 0 1 2 3 y 4 1 2 4 3 338 Chapter Expoetial Fuctios ad Sequeces