MIXED REVIEW of Problem Solvig STATE TEST PRACTICE classzoe.com Lessos 2.4 2.. MULTI-STEP PROBLEM A ball is dropped from a height of 2 feet. Each time the ball hits the groud, it bouces to 70% of its previous height.. SHORT RESPONSE Why does the sum of a ifiite geometric series ot exist if r where r is the commo ratio?. SHORT RESPONSE The legth l of the first loop of a sprig is iches. The legth l 2 of the secod loop is 0.9 times the legth of the first loop. The legth l 3 of the third loop is 0.9 times the legth of the secod loop, ad so o. If the sprig could have ifiitely may loops, would its legth be fiite or ifiite? Explai. If its legth is fiite, fid the legth. a. Write a ifiite series to model the total distace traveled by the ball, excludig the distace traveled before the first bouce. b. Fid the total distace traveled by the ball, icludig the distace traveled before the first bouce. 2. MULTI-STEP PROBLEM A fractal tree starts with a sigle brach (the truk). At each stage, the ew braches from the previous stage each grow two more braches as show. Stage Stage 2 Stage 3 a. List the umber of ew braches i each of the first six stages. b. Is the sequece of umbers from part (a) arithmetic, geometric, or either? c. Write a explicit rule ad a recursive rule for the sequece from part (a). 3. GRIDDED ANSWER What is the sum of the first three iterates of the fuctio f(x) x 2 2 8 whe the iitial value is x 0 2? 4. OPEN-ENDED Give a example of a explicit rule for a sequece ad a recursive rule for the same sequece.... 7. EXTENDED RESPONSE You take out a five year loa of $0,000 to buy a car. The loa has a aual iterest rate of.% compouded mothly. Each moth you make a mothly paymet of $9 (except the last moth whe you make a paymet of oly $). a. Fid the mothly iterest rate. The write a recursive rule for the amout of moey you owe after moths. b. How much moey do you owe after 2 moths? c. Suppose you had decided to pay a additioal $0 with each mothly paymet. Use a graphig calculator to fid the umber of moths you would have eeded to repay the loa. d. I your opiio, is it beeficial to pay the additioal $0 with each paymet? Explai your reasoig. 8. GRIDDED ANSWER A tree farm iitially has 8000 trees. Each year 0% of the trees are harvested ad 00 seedligs are plated. What umber of trees evetually exists o the farm after a exteded period of time? 9. OPEN-ENDED Write a ifiite geometric series that has a sum of 4. 838 Chapter 2 Sequeces ad Series 2pe-20.idd 838 0/2/0 2:0: PM
2 Big Idea CHAPTER SUMMARY BIG IDEAS Aalyze Sequeces For Your Notebook The iformatio below highlights the similarities ad differeces betwee arithmetic ad geometric sequeces. Arithmetic Sequece a a a ( 2 )d First term: a Geometric Sequece a a a r 2 First term: a Commo differece: d Commo ratio: r Graph is liear. Graph is expoetial. Big Idea 2 Fid Sums of Series The most commo formulas for sums of series are show below. Arithmetic Series Geometric Series Ifiite Geometric Series Sum of the first terms: S a a 2 2 Example: 4 9 4 9 24 S 4 24 2 2 70 Sum of the first terms: S a 2 r 2 r 2, r Þ Example: 3 2 24 S 4 3 2 24 2 2 2 4 Sum of the series: S a 2 r, r < Example: 0.2 0.04... S 2 0.2.2 Other commo sum formulas: ( ) i 2 i 2 ( )(2 ) Big Idea 3 Use Recursive Rules The table shows explicit ad recursive rules for arithmetic ad geometric sequeces. Arithmetic Sequece Example: 3,, 7, 9,,... Explicit Rule a a ( 2 )d a 2 Recursive Rule a a 2 d a 3, a a 2 2 Geometric Sequece Example: 8, 4, 2,, 0.,... a a r 2 a 8(0.) 2 a r p a 2 a 8, a 0.a 2 Chapter Summary 839 2pe-280.idd 839 0/2/0 2:0: PM
2 CHAPTER REVIEW classzoe.com REVIEW KEY VOCABULARY Multi-Laguage Glossary Vocabulary practice sequece, p. 794 terms of a sequece, p. 794 series, p. 79 summatio otatio, p. 79 sigma otatio, p. 79 arithmetic sequece, p. 802 commo differece, p. 802 arithmetic series, p. 804 geometric sequece, p. 80 commo ratio, p. 80 geometric series, p. 82 partial sum, p. 820 explicit rule, p. 827 recursive rule, p. 827 iteratio, p. 830 VOCABULARY. Copy ad complete: The values i the rage of a sequece are called the? of the sequece. 2. WRITING How ca you determie whether a sequece is arithmetic? 3. Copy ad complete: A()? rule gives a as a fuctio of the term s positio umber i the sequece. 4. Copy ad complete: I a()? sequece, the ratio of ay term to the previous term is costat. REVIEW AND Use the review examples ad exercises below to check your uderstadig of the cocepts you have leared i each lesso of Chapter 2. 2. Defie ad Use Sequeces ad Series pp. 794 800 Fid the sum of the series a 2 2 4 23 a 2 2 2 2 4 0 a 3 3 2 2 4 a 4 4 2 2 4 2 The sum of the series is 4 First term (i 2 2 4). Secod term Third term Fourth term 4 (i 2 2 4) 23 0 2 4. ad o p. 797 for Exs. 8 Fid the sum of the series.. ( 2 7). i 2 (0 2 4i) 7. 7 i 8. 2 k 2 k 840 Chapter 2 Sequeces ad Series 2pe-280.idd 840 0/2/0 2:0: PM
classzoe.com Chapter Review Practice 2.2 Aalyze Arithmetic Sequeces ad Series pp. 802 809 Write a rule for the th term of the sequece 9, 3, 7, 2, 2,.... The sequece is arithmetic with first term a 9 ad commo differece d 4. So, a rule for the th term is: a a ( 2 )d Write geeral rule. 9 ( 2 )(4) Substitute 9 for a ad 4 for d. 4 Simplify. 2, 3, 4, ad o pp. 803 80 for Exs. 9 Write a rule for the th term of the arithmetic sequece. 9. 8,, 2, 2, 24,... 0. d 7, a 8 4. a 4 27, a 9 Fid the sum of the series. 2. (3 2i) 3. 2 (2 2 3i) 4. 22 (i 2 ). 30 (284 8i). COMPUTER Joe buys a $00 computer o layaway by makig a $200 dow paymet ad the payig $2 per moth. Write a rule for the total amout of moey paid o the computer after moths. 2.3 Aalyze Geometric Sequeces ad Series pp. 80 87 Fid the sum of the series 7 (3) i 2. The series is geometric with first term a ad commo ratio r 3. S 7 a 2 r7 2 r 2 Write rule for S 7. 2 37 2 3 2 Substitute for a ad 3 for r. 4 Simplify. 2, 3, 4, ad o pp. 8 83 for Exs. 7 23 Write a rule for the th term of the geometric sequece. 7. 2, 4,, 4,,... 8. r, a 2 200 9. a 44, a 3 Fid the sum of the series. 20. 3() i 2 2. 9 8(2) i 2 22. 2 3 2 i 2 23. 7 40 2 2 i 2 Chapter Review 84 2pe-280.idd 84 0/2/0 2:0:8 PM
2 Fid 2.4 CHAPTER REVIEW Sums of Ifiite Geometric Series pp. 820 82 Fid the sum of the series 4 2 i 2, if it exists. For this series, a ad r 4. Because r <, the sum of this series exists. a The sum is S 2 r 2 4. 2 ad o pp. 82 822 for Exs. 24 3 Fid the sum of the ifiite geometric series, if it exists. 24. 3 8 2 i 2 2. 7 2 3 4 2 i 2 2. 4(.3) i 2 27. 20.2(0.) i 2 Write the repeatig decimal as a fractio i lowest terms. 28. 0.888... 29. 0.444... 30. 0.3787878... 3. 0.7838383... 2. Use Recursive Rules with Sequeces ad Fuctios pp. 827 833 Write a recursive rule for the sequece, 0, 4, 8, 22,.... The sequece is arithmetic with first term a ad commo differece d 0 2 4. a a 2 d Geeral recursive rule for a a 2 4 Substitute 4 for d. So, a recursive rule for the sequece is a, a a 2 4., 2, ad 3 o pp. 827 828 for Exs. 32 38 Write the first five terms of the sequece. 32. a 4, a a 2 9 33. a 8, a a 2 34. a 2, a p a 2 Write a recursive rule for the sequece. 3., 8, 4, 2, 48,... 3. 4,, 9, 3, 8,... 37. 7, 3, 9, 2, 3,... 38. POPULATION A tow s populatio icreases at a rate of about % per year. I 2000, the tow had a populatio of 2,000. Write a recursive rule for the tow s populatio P i year. Let represet 2000. 842 Chapter 2 Sequeces ad Series 2pe-280.idd 842 0/2/0 2:07:02 PM
2 Tell CHAPTER TEST whether the sequece is arithmetic, geometric, or either. Explai.., 9, 3, 7,... 2. 3,, 2, 24,... 3. 40, 0, 2,,... 4. 4, 7, 2, 9,... 8 Write the first six terms of the sequece.. a 2 2. a 7 3 7. a 4 8. a 2 a a 2 a a 2 Write the ext term of the sequece, ad the write a rule for the th term. 9.,, 7, 23,... 0. 3,, 7, 37,...., 7 0, 8, 9,... 2.., 3.2, 4.8,.4,... 20 Fid the sum of the series. 3. 48 i 4. 28 2. 0 (4i 2 9). 9 (2i ) 7. 9(2) i 2 8. 2 3 2 i 2 9. 8 3 4 2 i 2 20. 20 3 0 2 i 2 Write the repeatig decimal as a fractio i lowest terms. 2. 0.... 22. 0.444... 23. 0.878787... 24. 0.3222... Write a recursive rule for the sequece. 2. 2, 2, 72, 432,... 2. 3, 0, 7, 24,... 27. 3, 4,,,... 28., 23, 9, 227,... Fid the first three iterates of the fuctio for the give iitial value. 29. f(x) 3x 2 7, x 0 4 30. f(x) 8 2 x, x 0 3. f(x) x 2 2, x 0 2 32. QUILTS Use the patter of checkerboard quilts show., a 2, a 2 3, a 4, a 8 a. What does represet for each quilt? What does a represet? b. Make a table that shows ad a for, 2, 3, 4,,, 7, ad 8. c. Use the rule a 2 2 4 [ 2 (2) ] to fid a for, 2, 3, 4,,, 7, ad 8. Compare these values with the results i your table. What ca you coclude about the sequece defied by this rule? 33. AUDITIONS Several rouds of auditios are beig held to cast the three mai parts i a play. There are 3072 actors at the first roud of auditios. I each successive roud of auditios, oe fourth of the actors from the previous roud remai. Fid a rule for the umber a of actors i the th roud of auditios. For what values of does your rule make sese? Chapter Test 843 2pe-280.idd 843 0/2/0 2:07:0 PM