. Aalyzig Arithmetic Sequeces ad Series COMMON CORE Learig Stadards HSF-IF.A.3 HSF-BF.A. HSF-LE.A. Essetial Questio How ca you recogize a arithmetic sequece from its graph? I a arithmetic sequece, the differece of cosecutive terms, called the commo differece, is costat. For example, i the arithmetic sequece 1,, 7, 10,..., the commo differece is 3. Recogizig Graphs of Arithmetic Sequeces Work with a parter. Determie whether each graph shows a arithmetic sequece. If it does, the write a rule for the th term of the sequece, ad use a spreadsheet to fid the sum of the first 0 terms. What do you otice about the graph of a arithmetic sequece? a. 16 1 b. 16 1 6 6 c. 16 d. 16 1 1 6 6 Fidig the Sum of a Arithmetic Sequece REASONING ABSTRACTLY To be proficiet i math, you eed to make sese of quatities ad their relatioships i problem situatios. Work with a parter. A teacher of Germa mathematicia Carl Friedrich Gauss (1777 155) asked him to fid the sum of all the whole umbers from 1 through 100. To the astoishmet of his teacher, Gauss came up with the aswer after oly a few momets. Here is what Gauss did: 1 + + 3 +... + 100 100 + 99 + 9 +... + 1 101 + 101 + 101 +... + 101 100 101 = 5050 Explai Gauss s thought process. The write a formula for the sum S of the first terms of a arithmetic sequece. Verify your formula by fidig the sums of the first 0 terms of the arithmetic sequeces i Exploratio 1. Compare your aswers to those you obtaied usig a spreadsheet. Commuicate Your Aswer 3. How ca you recogize a arithmetic sequece from its graph?. Fid the sum of the terms of each arithmetic sequece. a. 1,, 7, 10,..., 301 b. 1,, 3,,..., 1000 c.,, 6,,..., 00 Sectio. Aalyzig Arithmetic Sequeces ad Series 17
. Lesso What You Will Lear Core Vocabulary arithmetic sequece, p. 1 commo differece, p. 1 arithmetic series, p. 0 Previous liear fuctio mea Idetify arithmetic sequeces. Write rules for arithmetic sequeces. Fid sums of fiite arithmetic series. Idetifyig Arithmetic Sequeces I a arithmetic sequece, the differece of cosecutive terms is costat. This costat differece is called the commo differece ad is deoted by d. Idetifyig Arithmetic Sequeces Tell whether each sequece is arithmetic. a. 9,, 5, 1, 19,... b. 3, 15, 9, 5, 3,... Fid the differeces of cosecutive terms. a. a a 1 = ( 9) = 7 a 3 a = 5 ( ) = 7 a a 3 = 1 5 = 7 a 5 a = 19 1 = 7 Each differece is 7, so the sequece is arithmetic. b. a a 1 = 15 3 = a 3 a = 9 15 = 6 a a 3 = 5 9 = a 5 a = 3 5 = The differeces are ot costat, so the sequece is ot arithmetic. Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com Tell whether the sequece is arithmetic. Explai your reasoig. 1., 5,, 11, 1,.... 15, 9, 3, 3, 9,... 3.,,, 1, 1,... Writig Rules for Arithmetic Sequeces Core Cocept Rule for a Arithmetic Sequece Algebra The th term of a arithmetic sequece with first term a 1 ad commo differece d is give by: = a 1 + ( 1)d Example The th term of a arithmetic sequece with a first term of 3 ad a commo differece of is give by: = 3 + ( 1), or = + 1 1 Chapter Sequeces ad Series
Writig a Rule for the th Term COMMON ERROR I the geeral rule for a arithmetic sequece, ote that the commo differece d is multiplied by 1, ot. Write a rule for the th term of each sequece. The fid a 15. a. 3,, 13, 1,... b. 55, 7, 39, 31,... a. The sequece is arithmetic with first term a 1 = 3, ad commo differece d = 3 = 5. So, a rule for the th term is = a 1 + ( 1)d Write geeral rule. = 3 + ( 1)5 Substitute 3 for a 1 ad 5 for d. = 5. Simplify. A rule is = 5, ad the 15th term is a 15 = 5(15) = 73. b. The sequece is arithmetic with first term a 1 = 55, ad commo differece d = 7 55 =. So, a rule for the th term is = a 1 + ( 1)d Write geeral rule. = 55 + ( 1)( ) Substitute 55 for a 1 ad for d. = + 63. Simplify. A rule is = + 63, ad the 15th term is a 15 = (15) + 63 = 57. Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com. Write a rule for the th term of the sequece 7, 11, 15, 19,.... The fid a 15. Writig a Rule Give a Term ad Commo Differece Oe term of a arithmetic sequece is a 19 = 5. The commo differece is d = 3. Write a rule for the th term. The graph the first six terms of the sequece. Step 1 Use the geeral rule to fid the first term. = a 1 + ( 1)d Write geeral rule. ANALYZING RELATIONSHIPS Notice that the poits lie o a lie. This is true for ay arithmetic sequece. So, a arithmetic sequece is a liear fuctio whose domai is a subset of the itegers. You ca also use fuctio otatio to write sequeces: a 19 = a 1 + (19 1)d Substitute 19 for. 5 = a 1 + 1( 3) Substitute 5 for a 19 ad 3 for d. 9 = a 1 Solve for a 1. Step Write a rule for the th term. = a 1 + ( 1)d Write geeral rule. = 9 + ( 1)( 3) Substitute 9 for a 1 ad 3 for d. = 3 + 1 Simplify. Step 3 Use the rule to create a table of values for the sequece. The plot the poits. 6 f () = 3 + 1. 1 3 5 6 9 6 3 0 3 6 6 Sectio. Aalyzig Arithmetic Sequeces ad Series 19
Writig a Rule Give Two Terms Two terms of a arithmetic sequece are a 7 = 17 ad a 6 = 93. Write a rule for the th term. Step 1 Write a system of equatios usig = a 1 + ( 1)d. Substitute 6 for to write Equatio 1. Substitute 7 for to write Equatio. a 6 = a 1 + (6 1)d 93 = a 1 + 5d Equatio 1 a 7 = a 1 + (7 1)d 17 = a 1 + 6d Equatio Step Solve the system. 76 = 19d Subtract. Check Use the rule to verify that the 7th term is 17 ad the 6th term is 93. a 7 = (7) 11 = 17 a 6 = (6) 11 = 93 = d Solve for d. 93 = a 1 + 5() Substitute for d i Equatio 1. 7 = a 1 Solve for a 1. Step 3 Write a rule for. = a 1 + ( 1)d Write geeral rule. = 7 + ( 1) Substitute for a 1 ad d. = 11 Simplify. Moitorig Progress Help i Eglish ad Spaish at BigIdeasMath.com Write a rule for the th term of the sequece. The graph the first six terms of the sequece. 5. a 11 = 50, d = 7 6. a 7 = 71, a 16 = 6 Fidig Sums of Fiite Arithmetic Series The expressio formed by addig the terms of a arithmetic sequece is called a arithmetic series. The sum of the first terms of a arithmetic series is deoted by S. To fid a rule for S, you ca write S i two differet ways ad add the results. S = a 1 + (a 1 + d ) + (a 1 + d ) +... + S = + ( d ) + ( d ) +... + a 1 S = (a 1 + ) + (a 1 + ) + (a 1 + ) +... + (a 1 + ) (a 1 + ) is added times. You ca coclude that S = (a 1 + ), which leads to the followig result. Core Cocept The Sum of a Fiite Arithmetic Series The sum of the first terms of a arithmetic series is S = ( a 1 + ). I words, S is the mea of the first ad th terms, multiplied by the umber of terms. 0 Chapter Sequeces ad Series
Fidig the Sum of a Arithmetic Series STUDY TIP This sum is actually a partial sum. You caot fid the complete sum of a ifiite arithmetic series because its terms cotiue idefiitely. 0 Fid the sum (3i + 7). Step 1 Fid the first ad last terms. Step Fid the sum. a 1 = 3(1) + 7 = 10 a 0 = 3(0) + 7 = 67 Idetify first term. Idetify last term. S 0 = 0 ( a 1 + a 0 ) Write rule for S 0. 10 + 67 = 0 ( ) Substitute 10 for a 1 ad 67 for a 0. = 770 Simplify. Solvig a Real-Life Problem You are makig a house of cards similar to the oe show. a. Write a rule for the umber of cards i the th row whe the top row is row 1. b. How may cards do you eed to make a house of cards with 1 rows? first row Check Use a graphig calculator to check the sum. sum(seq(3x,x,1,1 )) 3 a. Startig with the top row, the umber of cards i the rows are 3, 6, 9, 1,.... These umbers form a arithmetic sequece with a first term of 3 ad a commo differece of 3. So, a rule for the sequece is: = a 1 + ( 1)d Write geeral rule. = 3 + ( 1)(3) Substitute 3 for a 1 ad 3 for d. = 3 Simplify. b. Fid the sum of a arithmetic series with first term a 1 = 3 ad last term a 1 = 3(1) = 36. S 1 = 1 ( a 1 + a 1 ) = 1 ( 3 + 36 ) = 3 So, you eed 3 cards to make a house of cards with 1 rows. Moitorig Progress Fid the sum. 10 7. 9i Help i Eglish ad Spaish at BigIdeasMath.com 1. (7k + ) 9. ( + 6) k =1 =1 0 10. WHAT IF? I Example 6, how may cards do you eed to make a house of cards with eight rows? Sectio. Aalyzig Arithmetic Sequeces ad Series 1
. Exercises Dyamic Solutios available at BigIdeasMath.com Vocabulary ad Core Cocept Check 1. COMPLETE THE SENTENCE The costat differece betwee cosecutive terms of a arithmetic sequece is called the.. DIFFERENT WORDS, SAME QUESTION Which is differet? Fid both aswers. What sequece cosists of all the positive odd umbers? What sequece starts with 1 ad has a commo differece of? What sequece has a th term of = 1 + ( 1)? What sequece has a th term of = + 1? Moitorig Progress ad Modelig with Mathematics I Exercises 3 10, tell whether the sequece is arithmetic. Explai your reasoig. (See Example 1.) 3. 1, 1, 3, 5, 7,.... 1, 6, 0, 6, 1,... 5. 5,, 13, 0, 9,... 6. 3, 5, 9, 15, 3,... 7. 36, 1, 9, 9, 9,.... 1, 7, 9, 3, 1,... 9. 1, 3, 1, 5, 3,... 10. 1 6, 1, 5 6, 7 6, 3,... 11. WRITING EQUATIONS Write a rule for the arithmetic sequece with the give descriptio. a. The first term is 3 ad each term is 6 less tha the previous term. b. The first term is 7 ad each term is 5 more tha the previous term. 1. WRITING Compare the terms of a arithmetic sequece whe d > 0 to whe d < 0. I Exercises 13 0, write a rule for the th term of the sequece. The fid a 0. (See Example.) 13. 1, 0,, 36,... 1. 7, 1, 17,,... 15. 51,, 5,,... 16. 6, 79, 7, 65,... 17. 1, 1 3, 1 3, 1,... 1., 5, 1, 1,... 19..3, 1.5, 0.7, 0.1,... 0. 11.7, 10., 9.9, 9,... ERROR ANALYSIS I Exercises 1 ad, describe ad correct the error i writig a rule for the th term of the arithmetic sequece, 9,, 17, 30,.... 1.. Use a 1 = ad d = 13. = a 1 + d = + ( 13) = 13 The first term is ad the commo differece is 13. = 13 + ( 1)() = 35 + I Exercises 3, write a rule for the th term of the sequece. The graph the first six terms of the sequece. (See Example 3.) 3. a 11 = 3, d = 5. a 13 =, d = 5. a 0 = 7, d = 6. a 15 = 35, d = 3 1 7. a 17 = 5, d =. a 1 = 5, d = 3 9. USING EQUATIONS Oe term of a arithmetic sequece is a = 13. The commo differece is. What is a rule for the th term of the sequece? A = 51 + B = 35 + C = 51 D = 35 Chapter Sequeces ad Series
30. FINDING A PATTERN Oe term of a arithmetic sequece is a 1 = 3. The commo differece is 6. What is aother term of the sequece? A a 3 = 11 B a = 53 C a 5 = 13 D a 6 = 7 I Exercises 31 3, write a rule for the th term of the arithmetic sequece. (See Example.) 31. a 5 = 1, a 10 = 96 3. a 7 = 5, a 11 = 9 33. a 6 =, a 15 = 6 3. a = 15, a 17 = 7 35. a 1 = 59, a 1 = 71 36. a 1 = 3, a 19 = 73 37. a = 1, a 16 = 3. a 1 = 9, a 7 = 15 WRITING EQUATIONS I Exercises 39, write a rule for the sequece with the give terms. 39. (1, 9) (, 6) (3, 3) (, 0) 0. 0 10 (1, 15) (, 10) (3, 5) (, 0) 6. DRAWING CONCLUSIONS Describe how doublig each term i a arithmetic sequece chages the commo differece of the sequece. Justify your aswer. I Exercises 7 5, fid the sum. (See Example 5.) 0 7. (i 3). (i + 7) 33 9. (6 i ) 50. ( 3 i ) 6 1 51. (.3 + 0.1i ) 5. (.1 + 0.i ) NUMBER SENSE I Exercises 53 ad 5, fid the sum of the arithmetic sequece. 53. The first 19 terms of the sequece 9,, 5, 1,.... 5. The first terms of the sequece 17, 9, 1, 7,.... 55. MODELING WITH MATHEMATICS A marchig bad is arraged i rows. The first row has three bad members, ad each row after the first has two more bad members tha the row before it. (See Example 6.) a. Write a rule for the umber of bad members i the th row. b. How may bad members are i a formatio with seve rows? 31 39 1. 3. (, 5) (3, ) 1 3 (, 1) (1, ). 1 6 6 5 6 7 5 9 33 37 1 (, 16) (3, 9) (, ) (1, 5) 56. MODELING WITH MATHEMATICS Domestic bees make their hoeycomb by startig with a sigle hexagoal cell, the formig rig after rig of hexagoal cells aroud the iitial cell, as show. The umber of cells i successive rigs forms a arithmetic sequece.. 5 6 7 31 39 7 55 63 Iitial cell 1 rig rigs 5. WRITING Compare the graph of = 3 + 1, where is a positive iteger, with the graph of f (x) = 3x + 1, where x is a real umber. a. Write a rule for the umber of cells i the th rig. b. How may cells are i the hoeycomb after the ith rig is formed? Sectio. Aalyzig Arithmetic Sequeces ad Series 3
57. MATHEMATICAL CONNECTIONS A quilt is made up of strips of cloth, startig with a ier square surrouded by rectagles to form successively larger squares. The ier square ad all rectagles have a width of 1 foot. Write a expressio usig summatio otatio that gives the sum of the areas of all the strips of cloth used to make the quilt show. The evaluate the expressio. 60. THOUGHT PROVOKING I umber theory, the Dirichlet Prime Number Theorem states that if a ad b are relatively prime, the the arithmetic sequece a, a + b, a + b, a + 3b,... cotais ifiitely may prime umbers. Fid the first 10 primes i the sequece whe a = 3 ad b =. 61. REASONING Fid the sum of the positive odd itegers less tha 300. Explai your reasoig. 6. USING EQUATIONS Fid the value of. 5. HOW DO YOU SEE IT? Which graph(s) represets a arithmetic sequece? Explai your reasoig. a. c. 6 6 6 b. d. 1 6 59. MAKING AN ARGUMENT Your fried believes the sum of a series doubles whe the commo differece of a arithmetic series is doubled ad the first term ad umber of terms i the series remai uchaged. Is your fried correct? Explai your reasoig. a. (3i + 5) = 5 b. ( i 1) = 117 c. (7 + 1i) = 55 d. ( 3 i) = 507 i =5 i =3 63. ABSTRACT REASONING A theater has rows of seats, ad each row has d more seats tha the row i frot of it. There are x seats i the last (th) row ad a total of y seats i the etire theater. How may seats are i the frot row of the theater? Write your aswer i terms of, x, ad y. 6. CRITICAL THINKING The expressios 3 x, x, ad 1 3x are the first three terms i a arithmetic sequece. Fid the value of x ad the ext term i the sequece. 65. CRITICAL THINKING Oe of the major sources of our kowledge of Egyptia mathematics is the Ahmes papyrus, which is a scroll copied i 1650 B.C. by a Egyptia scribe. The followig problem is from the Ahmes papyrus. Divide 10 hekats of barley amog 10 me so that the commo differece is 1 of a hekat of barley. Use what you kow about arithmetic sequeces ad series to determie what portio of a hekat each ma should receive. Maitaiig Mathematical Proficiecy Simplify the expressio. (Sectio 5.) Reviewig what you leared i previous grades ad lessos 7 66. 7 1/3 67. 3 3 6. ( 9) 9 1/ 69. (5 1/ 5 1/ ) Tell whether the fuctio represets expoetial growth or expoetial decay. The graph the fuctio. (Sectio 6.) 70. y = e x 71. y = e 3x 7. y = 3e x 73. y = e 0.5x Chapter Sequeces ad Series