SEQUENCES AND SERIES CHAPTER


 Randell Shaw
 3 years ago
 Views:
Transcription
1 CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each moth was % of the upaid balacethe amout that the Gat family still owed after each paymet is a fuctio of the umber of moths that have passed sice they purchased the computer At the ed of moth : they owed (,200, ) dollars or (, ) $,8 At the ed of moth 2: they owed (,8, ) dollars or (, ) $,03477 Each moth, iterest is added to the balace from the previous moth ad a paymet of $00 is subtracted We ca express the mothly paymets with the fuctio defied as {(, f())} Let the domai be the set of positive itegers that represet the umber of moths sice the iitial purchasethe: f(),8 f(2),03477 I geeral, for positive itegers : f() f( 2 ) This patter cotiues util (f( 2 ) 3 0) is betwee 0 ad 00, sice the fial paymet would be (f( 2 ) 3 0) dollars I this chapter we will study sequetial fuctios, such as the oe described here, whose domai is the set of positive itegers 6 CHAPTER TABLE OF CONTENTS 6 Sequeces 62 Arithmetic Sequeces 63 Sigma Notatio 64 Arithmetic Series 6 Geometric Sequeces 66 Geometric Series 67 Ifiite Series Chapter Summary Vocabulary Review Exercises Cumulative Review 247
2 248 Sequeces ad Series 6 SEQUENCES A ball is dropped from height of 6 feet Each time that it bouces, it reaches a height that is half of its previous height We ca list the height to which the ball bouces i order util it fially comes to rest After Bouce Height (ft) The umbers 8, 4, 2,, 2, 4, 8 form a sequecea sequece is a set of umbers writte i a give order We ca list these heights as ordered pairs of umbers i which each height is paired with the umber that idicates its positio i the list The set of ordered pairs would be: {(, 8), (2, 4), (3, 2), (4, ), (, 0), (6, 02), (7, 02)} We associate each term of the sequece with the positive iteger that specifies its positio i the ordered set Therefore, a sequece is a special type of fuctio DEFINITION A fiite sequece is a fuctio whose domai is the set of itegers {, 2, 3,, } y The fuctio that lists the height of the ball after 7 bouces is show o the graph at the right Ofte the sequece ca cotiue without ed I this case, the domai is the set of positive itegers O x DEFINITION A ifiite sequece is a fuctio whose domai is the set of positive itegers The terms of a sequece are ofte desigated as a, a 2, a 3, a 4, a, If the sequece is desigated as the fuctio f, the f() a, f(2) a 2, or i geeral: f() a Most sequeces are sets of umbers that are related by some patter that ca be expressed as a formula The formula that allows ay term of a sequece, except the first, to be computed from the previous term is called a recursive defiitio
3 For example, the sequece that lists the heights to which a ball bouces whe dropped from a height of 6 feet is 8, 4, 2,, 0, 02, 02, I this sequece, each term after the first is 2 the previous term Therefore, for each term after the first, 4 2 (8),2 2 (4), 2 (2), 0 2 (), 02 2 (0), 02 2 (02), For, we ca write the recursive defiitio: a 2 a 2 Alteratively, we ca write the recursive defiitio as: Sequeces 249 a 2 a for $ A rule that desigates ay term of a sequece ca ofte be determied from the first few terms of the sequece EXAMPLE a List the ext three terms of the sequece 2, 4, 8, 6, b Write a geeral expressio for a c Write a recursive defiitio for the sequece Solutio a It appears that each term of the sequece is a power of 2: 2,2 2,2 3,2 4, Therefore, the ext three terms should be 2,2 6, ad 2 7 or 32, 64, ad 28 b Each term is a power of 2 with the expoet equal to the umber of the term Therefore, a 2 c Each term is twice the previous term Therefore, for, a 2a 2 Alteratively, for $, a 2a Aswers a 32, 64, 28 b a 2 c For, a 2a 2 or for $, a 2a EXAMPLE 2 Write the first three terms of the sequece a 3 2 Solutio a 3() 2 2 a 2 3(2) 2 a 3 3(3) 2 8 Aswer The first three terms of the sequece are 2,, ad 8
4 20 Sequeces ad Series Sequeces ad the Graphig Calculator We ca use the sequece fuctio o the graphig calculator to view a sequece for a specific rage of terms Evaluate a sequece i terms of variable from a begiig term to a edig term That is: seq(sequece, variable, begiig term, edig term) For example, we ca examie the sequece a 2 2 o the calculator To view the first 20 terms of the sequece: ENTER: 2d LIST X,T,, x 2 2 X,T,,,, 20 ) ENTER, DISPLAY: seq(x 2 +2, X,, 20) { Note: We use the variable X istead of to eter the sequece We ca also use the left ad right arrow keys to examie the terms of the sequece Exercises Writig About Mathematics Nichelle said that sequece of umbers i which each term equals half of the previous term is a fiite sequece Radi said that it is a ifiite sequece Who is correct? Justify your aswer 2 a Jacob said that if a 3 2, the a a 3 Do you agree with Jacob? Explai why or why ot b Carlos said that if a 2, the a 2 Do you agree with Carlos? Explai why or why ot Developig Skills I 3 8, write the first five terms of each sequece 3 a 4 a a 2 6 a 7 a 2 8 a a 3 0 a 2 a a a 2 4 a a 2 6 a a 3 8 a 2 i
5 I 9 30: a Write a algebraic expressio that represets a for each sequece b Fid the ith term of each sequece 9 2, 4, 6, 8, 20 3, 6, 9, 2, 2, 4, 7, 0, 22 3, 9, 27, 8, 23 2, 6, 3,, 24 7, 9,, 3, i,8i,6i,4i, 26 2, 3, 4,, 27 2, 3, 4,, 28 2,, 0, 7, 29, 22, 3, 24, 30,!2,!3,2, I 3 39, write the first five terms of each sequece 3 a, a = a 2 32 a, a 3a 33 a, a 2a 2 34 a 22, a 22a 2 3 a 20, a a a 4, a a 37 a 2 36, a 3 a 2 38 a 3 2, a 2a 39 a 2, a a 2 Applyig Skills Sequeces 2 40 Sea has started a exercise program The first day he worked out for 30 miutes Each day for the ext six days, he icreased his time by miutes a Write the sequece for the umber of miutes that Sea worked out for each of the seve days b Write a recursive defiitio for this sequece 4 Sherri wats to icrease her vocabulary O Moday she leared the meaigs of four ew words Each other day that week, she icreased the umber of ew words that she leared by two a Write the sequece for the umber of ew words that Sherri leared each day for a week b Write a recursive defiitio for this sequece 42 Julie is tryig to lose weight She ow weighs 80 pouds Every week for eight weeks, she was able to lose 2 pouds a List Julie s weight for each week b Write a recursive defiitio for this sequece 43 Jauary, 2008, was a Tuesday a List the dates for each Tuesday i Jauary of that year b Write a recursive defiitio for this sequece 44 Hui started a ew job with a weekly salary of $400 After oe year, ad for each year that followed, his salary was icreased by 0% Hui left this job after six years a List the weekly salary that Hui eared each year b Write a recursive defiitio for this sequece 4 Oe of the most famous sequeces is the Fiboacci sequece I this sequece, a, a 2, ad for 2, a = a 22 a 2 Write the first te terms of this sequece
6 22 Sequeces ad Series HadsO Activity The Tower of Haoi is a famous problem that has challeged problem solvers throughout the ages The tower cosists of three pegs O oe peg there are a umber of disks of differet sizes, stacked accordig to size with the largest at the bottom The task is to move the etire stack from oe peg to the other side usig the followig rules: Oly oe disk may be moved at a time No disk may be placed o top of a smaller disk Note that a move cosists of takig the top disk from oe peg ad placig it o aother peg Use a stack of differetsized cois to model the Tower of Haoi What is the smallest umber of moves eeded if there are: a 2 disks? b 3 disks? c 4 disks? d disks? 2 Write a recursive defiitio for the sequece described i Exercise, a d 62 ARITHMETIC SEQUENCES The set of positive odd umbers,, 3,, 7,, is a sequece The first term, a, is ad each term is 2 greater tha the precedig term The differece betwee cosecutive terms is 2 We say that 2 is the commo differece for the sequece The set of positive odd umbers is a example a arithmetic sequece DEFINITION A arithmetic sequece is a sequece such that for all, there is a costat d such that a 2 a d For a arithmetic sequece, the recursive formula is: a a 2 d A arithmetic sequece is formed whe each term after the first is obtaied by addig the same costat to the previous term For example, look at the first five terms of the sequece of positive odd umbers
7 Arithmetic Sequeces 23 a a a 2 a a 2 (2) a 3 a a 3 [ (2)] 2 2(2) a 4 a a 4 [ 2(2)] 2 3(2) a a a [ 3(2)] 2 4(2) For this sequece, a ad each term after the first is foud by addig 2 to the precedig term Therefore, for each term, 2 has bee added to the first term oe less time tha the umber of the term For the secod term, 2 has bee added oce; for the third term, 2 has bee added twice; for the fourth term, 2 has bee added three times I geeral, for the th term, 2 has bee added 2 times Show below is a geeral arithmetic sequece with a as the first term ad d as the commo differece: a, a d, a 2d, a 3d, a 4d, a d,, a ( 2 )d If the first term of a arithmetic sequece is a ad the commo differece is d, the for each term of the sequece, d has bee added to a oe less time tha the umber of the term Therefore, ay term a of a arithmetic sequece ca be evaluated with the formula a a ( 2 )d where d is the commo differece of the sequece EXAMPLE For the arithmetic sequece 00, 97, 94, 9,, fid: a the commo differece b the 20th term of the sequece Solutio a The commo differece is the differece betwee ay term ad the previous term: or or The commo differece is 23 b a (20 2 )(23) 00 (9)(23) The 20th term ca also be foud by writig the sequece: 00, 97, 94, 9, 88, 8, 82, 79, 76, 73, 70, 67, 64, 6, 8,, 2, 49, 46, 43 Aswers a d 23 b a 20 43
8 24 Sequeces ad Series EXAMPLE 2 Scott is savig to buy a guitar I the first week, he put aside $42 that he received for his birthday, ad i each of the followig weeks, he added $8 to his savigs He eeds $400 for the guitar that he wats I which week will he have eough moey for the guitar? Solutio Let a 42 ad d 8 The value for for which a 400 is the week Scott will have eough moey to buy the guitar a a ( 2 )d ( 2 )(8) Scott adds moey to his savigs i $8 icremets, so he will ot have the eeded $400 i savigs util the 46th week Aswer 46th week EXAMPLE 3 The 4th term of a arithmetic sequece is 80 ad the 2th term is 32 a What is the commo differece? b What is the first term of the sequece? Solutio a How to Proceed () Use a a ( 2 )d to write two equatio i two variables: (2) Subtract to elimiate a : (3) Solve for d: b Substitute i either equatio to fid a 80 a 3(26) 80 a a Aswers a d 26 b a a (4 2 )d 80 a 3d 32 a (2 2 )d 2(32 a d) 48 28d 26 d
9 Arithmetic Sequeces 2 Arithmetic Meas We have defied the arithmetic mea of two umbers as their average, that is, the sum of the umbers divided by 2 For example, the arithmetic mea of 4 ad is The umbers 4, 6, ad 28 form a arithmetic sequece with a commo differece of 2 We ca fid three umbers betwee 4 ad 28 that together with 4 ad 28 form a arithmetic sequece: 4, a 2, a 3, a 4, 28 This is a sequece of five terms Use the formula for a to fid the commo differece: a a ( 2 )d 28 4 ( 2 )d d 24 4d 6 d 2 Now use the recursive formula, a a d, ad d 6, to write the sequece: 4, 0, 6, 22,28 The umbers 0, 6, ad 22 are three arithmetic meas betwee 4 ad 28 Lookig at it aother way, i ay arithmetic sequece, ay give term is the average of the term before it ad the term after it Thus, a arithmetic mea is ay term of a arithmetic sequece EXAMPLE 4 Solutio Fid five arithmetic meas betwee 2 ad 23 Five arithmetic meas betwee 2 ad 23 will form a sequece of seve terms Use the formula for a 7 to fid the commo differece: a 7 a ( 2 )d 23 2 (7 2 )d d 2 6d 7 2 d Evaluate a 2, a 3, a 4, a, ad a 6 : a 2 2 (2 2 ) 7 2 a a 3 2 (3 2 ) 7 2 a 3 9 a 4 2 (4 2 ) 7 2 a a 2 ( 2 ) 7 2 a 6 a 6 2 (6 2 ) 7 2 a Aswer The five arithmetic meas are 2, 9, 2 2, 6, ad 9 2
10 26 Sequeces ad Series Exercises Writig About Mathematics Virgiia said that Example 3 could have bee solved without usig equatios Sice there are eight terms from a 4 to a 2, the differece betwee 80 ad 32 has to be divided ito eight parts Each part is 6 Sice the sequece is decreasig, the commo differece is 26 The usig 26, work back from a 4 to a : 80, 86, 92, 98 Do you thik that Virgiia s solutio is better tha the oe give i Example 3? Explai why or why ot 2 Pedro said that to form a sequece of five terms that begis with 2 ad eds with 2, you should divide the differece betwee 2 ad 2 by to fid the commo differece Do you agree with Pedro? Explai why or why ot Developig Skills I 3 8, determie if each sequece is a arithmetic sequece If the sequece is arithmetic, fid the commo differece 3 2,,8,,4, 4 3i, 2i,i,3i,i,,, 2, 3,, 8, 6 20,, 0,, 0, 7, 2, 4, 8, 6, 8, 2,, 7, 2, I 9 4: a Fid the commo differece of each arithmetic sequece b Write the th term of each sequece for the give value of 9 3, 6, 9, 2,, 8 0 2, 7, 2, 7,, 2 8, 6, 4, 2,, ,, 2,2,, 7 3 2, 23, 2, 27,, 0 4 2, 22, 23, 24,, 20 Write the first six terms of the arithmetic sequece that has 2 for the first term ad 42 for the sixth term 6 Write the first ie terms of the arithmetic sequece that has 00 as the fifth term ad 80 as the ith term 7 Fid four arithmetic meas betwee 3 ad 8 8 Fid two arithmetic meas betwee ad 9 Write a recursive defiitio for a arithmetic sequece with a commo differece of 23 Applyig Skills 20 O July, Mr Taylor owed $6,000 O the st of each of the followig moths, he repaid $400 List the amout owed by Mr Taylor o the 2d of each moth startig with July 2 Explai why the amout owed each moth forms a arithmetic sequece 2 Li is developig a fitess program that icludes doig pushups each day O each day of the first week he did 20 pushups Each subsequet week, he icreased his daily pushups by Durig which week did he do 60 pushups a day? a Use a formula to fid the aswer to the questio b Write the arithmetic sequece to aswer the questio c Which method do you thik is better? Explai you aswer
11 Sigma Notatio a Show that a liear fuctio whose domai is the set of positive itegers is a arithmetic sequece b For the liear fuctio y mx b, y a ad x Express a ad d of the arithmetic sequece i terms of m ad b 23 Leslie oticed that the daily umber of messages she received over the course of two moths form a arithmetic sequece If she received 3 messages o day 3 ad 64 messages o day 20: a How may messages did Leslie receive o day 2? b How may messages will Leslie receive o day 0? 63 SIGMA NOTATION Ke wats to get more exercise so he begis by walkig for 20 miutes Each day for two weeks, he icreases the legth of time that he walks by miutes At the ed of two weeks, the legth of time that he has walked each day is give by the followig arithmetic sequece: 20, 2, 30, 3, 40, 4, 0,, 60, 6, 70, 7, 80, 8 The total legth of time that Ke has walked i two weeks is the sum of the terms of this arithmetic sequece: This sum is called a series DEFINITION A series is the idicated sum of the terms of a sequece The symbol S, which is the Greek letter sigma, is used to idicate a sumthe umber of miutes that Ke walked o the th day is a 20 ( 2 )() We ca write the sum of the umber of miutes that Ke walked i sigma otatio: The below S is the value of for the first term of the series, ad the umber above S is the value of for the last term of the series The symbol 4 a a 4 4 a a a 20 ( 2 )() ca be read as the sum of a for all itegral values of from to 4
12 28 Sequeces ad Series EXAMPLE I expaded form: a a 2 a 3 a 4 a a 6 a 7 a 8 a 9 a 0 a a 2 a 3 a 4 [20 0()] [20 ()] [20 2()] c a 20 ( 2 )() [20 3()] ???8 For example, we ca idicate the sum of the first 0 positive eve umbers as a 2i a 2i 2() 2(2) 2(3) 2(4)???2(0) ??? 00 Note that i this case, i was used to idicate the umber of the term Although ay variable ca be used,, i, ad k are the variables most frequetly used Be careful ot to cofuse the variable i with the imagiary umber i The series a 2i is a example of a fiite series sice it is the sum of a fiite umber of terms A ifiite series is the sum of a ifiite umber of terms of a sequece We idicate that a series is ifiite by usig the symbol for ifiity, ` For example, we ca idicate the sum of all of the positive eve umbers as: Write the sum give by a (k ) Solutio a (k ) ( ) (2 ) (3 ) (4 ) ( ) (6 ) (7 ) EXAMPLE 2 4 a a 4 7 k 0 i 0 i 0 i ` a 2i c 2i c i 7 k Aswer Write the sum of the first 2 positive odd umbers i sigma otatio Solutio The positive odd umbers are, 3,, 7, The st positive odd umber is less tha twice, the 2d positive odd umber is less tha twice 2, the 3rd positive odd umber is less tha twice 3 I geeral, the th positive odd umber is less tha twice or a 2 2 The sum of the first 2 odd umbers is a (2 2 ) Aswer 2
13 Sigma Notatio 29 EXAMPLE 3 Use sigma otatio to write the series i two differet ways: a Express each term as a sum of two umbers, oe of which is a square b Express each term as a product of two umbers Solutio a The terms of this series ca be writte as 3 2 3, 4 2 4, 2,,0 2 0, or, i geeral, as 2 with from 3 to 0 The series ca be writte as a ( 2 ) b Write the series as 3(4) 4() (6) 6(7) 7(8) 8(9) 9(0) 0() The series is the sum of ( ) from 3 to 0 The series ca be writte as a ( ) 0 Aswers a a ( 2 ) b a ( ) EXAMPLE 4 Use sigma otatio to write the sum of the reciprocals of the atural umbers Solutio The reciprocals of the atural umbers are, 2, 3, 4, c, Sice there is o largest atural umber, this sequece has o last term Therefore, the sum of the terms of this sequece is a ifiite series I sigma otatio, the sum of the reciprocals of the atural umbers is: ` a Aswer Fiite Series ad the Graphig Calculator The graphig calculator ca be used to fid the sum of a fiite serieswe use the sum( fuctio alog with the seq( fuctio of the previous sectio to evaluate a series For example, to evaluate a k(k 2) o the calculator: 37 k
14 260 Sequeces ad Series STEP Eter the sequece ito the calculator ad store it i list L ENTER: 2d LIST ( X,T,, ( X,T,, 2 ) ), X,T,,, 37 ) STO 2d L ENTER, DISPLAY: seq(/(x(x+2)), X,, 37) >L { STEP 2 Use the sum( fuctio to fid the sum ENTER: 2d LIST 2d L ENTER DISPLAY: sum(l The sum is approximately equal to 072 Exercises Writig About Mathematics Is the series give i Example 3 equal to a f( 2) 2 ( 2)g? Justify your aswer 2 Explai why a k is udefied Developig Skills I 3 4: a Write each arithmetic series as the sum of terms b Fid each sum 0 3 a 3 4 a (2k 2 2) a 6 6 a 3 7 a (00 2 k) 8 a (3 2 3) 9 a ( 2 2i) 0 a (2) h h a f4 2 ( )g a 2ki 3 a ( 2 4k) 4 a (22) k 0 k0 k 0 k 0 h k3 8 4 k 2 k 0 0
15 Sigma Notatio 26 I 26, write each series i sigma otatio c c c 26! 2! 3! 4!! c Applyig Skills 27 Show that a ka i k a a i 28 Show that a (a i b i ) a a i a b i 29 I a theater, there are 20 seats i the first row Each row has 3 more seats tha the row ahead of it There are 3 rows i the theater a Express the umber of seats i the th row of the theater i terms of b Use sigma otatio to represet the umber of seats i the theater 30 O Moday, Elaie spet 4 miutes doig homework O the remaiig four days of the school week, she spet miutes loger doig homework tha she had the day before a Express the umber of miutes Elaie spet doig homework o the th day of the school week b Use sigma otatio to represet the total umber of miutes Elaie spet doig homework from Moday to Friday 3 Use the graphig calculator to evaluate the followig series to the earest hudredth: () a A B (2) a k (3) a i i i i k i ( 2 )(2)
16 262 Sequeces ad Series 64 ARITHMETIC SERIES I the last sectio, we wrote the sequece of miutes that Ke walked each day for two weeks: 20, 2, 30, 3, 40, 4, 0,, 60, 6, 70, 7, 80, 8 Sice the differece betwee each pair of cosecutive times is a costat,, the sequece is a arithmetic sequece The total legth of time that Ke walked i two weeks is the sum of the terms of this sequece: I geeral, if a, a d,a 2d,,a ( 2 )d is a arithmetic sequece with terms, the: a fa (i 2 )dg a (a d) (a 2d) c fa ( 2 )dg i This sum is called a arithmetic series DEFINITION A arithmetic series is the idicated sum of the terms of a arithmetic sequece Oce a give series is defied, we ca refer to it simply as S (for sigma) S is called the th partial sum ad represets the sum of the first terms of the sequece We ca fid the umber of miutes that Ke walked i 4 days by addig the 4 umbers or by observig the patter of this series Begi by writig the sum first i the order give ad the i reverse order S S Note that for this arithmetic series: a a 4 a 2 a 3 = a 3 a 2 = a 4 a = a a 0 = a 6 a 9 a 7 a 8 Add the sums together, combiig correspodig terms The sum of each 4 pair is 0 ad there are 2 or 7 pairs S S S S 4 4(0) Write the expressio i factored form S 4 7(0) Divide both sides by 2 S 4 73 Simplify
17 EXAMPLE Therefore, the total umber of miutes that Ke walked i 4 days is 7(0) or 73 miutes Does a similar patter exist for every arithmetic series? Cosider the geeral arithmetic series with terms, a a ( 2 )d List the terms of the series i ascedig order from a to a ad i descedig order from a to a Ascedig Order Descedig Order Sum I geeral, for ay arithmetic series with terms there are 2 pairs of terms whose sum is a a : 2S (a a ) (2a ( 2 )d) S 2 (a a ) 2 (2a ( 2 )d) a Write the sum of the first terms of the arithmetic series i sigma otatio b Fid the sum Solutio a For the related arithmetic sequece,, 4, 7,, the commo differece is 3 Therefore, a ( 2 )(3) The series is writte as a Aswer b Use the formula S 2 (2a ( 2 )d) with a,, ad d 3 S 2 f2() ( 2 )(3)g 2 f2 4(3)g 2 f2 42g 2 f44g 330 Aswer Arithmetic Series 263 a a a ( 2 )d a [a ( 2 )d] 2a ( 2 )d a 2 a d a a ( 2 2)d [a d] [a ( 2 2)d] 2a ( 2 )d a 3 a 2d a 2 a ( 2 3)d [a 2d] [a ( 2 3)d] 2a ( 2 )d ( ( a 2 a ( 2 3)d a 3 a 2d [a ( 2 3)d] [a 2d] 2a ( 2 )d a a ( 2 2)d a 2 a d [a ( 2 2)d] [a d] 2a ( 2 )d a a ( 2 )d a [a ( 2 )d] a 2a ( 2 )d ( 4 7 c
18 264 Sequeces ad Series Note: Part b ca also be solved by usig the formula S 2 (a a ) First fid a a a ( 2 )d S 2 ( 43) ( 2 )(3) 2 (44) 4(3) Notice that for this series there are 7 2 pairs The first 7 umbers are paired with the last 7 umbers, ad each pair has a sum of 44 The middle umber, 22, is paired with itself, makig half of a pair EXAMPLE 2 The sum of the first ad the last terms of a arithmetic sequece is 80 ad the sum of all the terms is,200 How may terms are i the sequece? Solutio S 2 (a a ),200 2 (80), Aswer There are 30 terms i the sequece Exercises Writig About Mathematics Is there more tha oe arithmetic series such that the sum of the first ad the last terms is 80 ad the sum of the terms is,200? Justify your aswer 2 Is a arithmetic series? Justify your aswer Developig Skills I 3 8, fid the sum of each series usig the formula for the partial sum of a arithmetic series Be sure to show your work i 4i 8i 2i 6i 20i !2 2!2 3!2 4!2 c!2
19 I 9 8, use the give iformatio to a write the series i sigma otatio, ad b fid the sum of the first terms 9 a 3, a 39, d 4 0 a 24, a 0, 6 a 24, a 0, d 26 2 a 0, d 2, 4 3 a 2, d 2, 4 a 0, d 22, 0 a 3, d 3, 2 6 a 00, d 2, 20 7 a 0 0, d 2, 0 8 a, d 2, 2 I 9 24: a Write each arithmetic series as the sum of terms b Fid the sum Arithmetic Series a 2k 20 a (3 k) 2 k 8 k2 9 a (20 2 2) a (00 2 i) 23 a i0 2 0 a (2 2) Applyig Skills 2 Madelie is writig a computer program for class The first day she wrote lies of code ad each day, as she becomes more skilled i writig code, she writes oe more lie tha the previous day It takes Madelie 6 days to complete the program How may lies of code did she write? 26 Jose is learig to crosscoutry ski He bega by skiig mile the first day ad each day he icreased the distace skied by 02 mile util he reached his goal of 3 miles a How may days did it take Jose to reach his goal? b How may miles did he ski from the time he bega util the day he reached his goal? 27 Sarah wats to save for a special dress for the prom The first moth she saved $ ad each of the ext five moths she icreased the amout that she saved by $2 What is the total amout Sarah saved over the six moths? 28 I a theater, there are 20 seats i the first row Each row has 3 more seats tha the row ahead of it There are 3 rows i the theater Fid the total umber of seats i the theater 29 O Moday, Eid spet 4 miutes doig homework O the remaiig four days of the school week she spet miutes loger doig homework tha she had the day before Fid the total umber of miutes Eid spet doig homework from Moday to Friday 30 Keega started a job that paid $20,000 a year Each year after the first, he received a raise of $600 What was the total amout that Keega eared i six years? 3 A ew health food store s et icome was a loss of $2,300 i its first moth, but its et icome icreased by $7 i each succeedig moth for the ext year What is the store s et icome for the year?
20 266 Sequeces ad Series 6 GEOMETRIC SEQUENCES Pete reuses paper that is blak o oe side to write phoe messages Oe day he took a stack of five sheets of paper ad cut it ito three parts ad the cut each part ito three parts The umber of pieces of paper that he had after each cut forms a sequece:,, 4 This sequece ca be writte as:, (3), (3) 2 Each term of the sequece is formed by multiplyig the previous term by 3, or we could say that the ratio of each term to the previous term is a costat, 3 If Pete had cotiued to cut the pieces of paper i thirds, the terms of the sequece would be, (3), (3) 2, (3) 3, (3) 4, (3), This sequece is called a geometric sequece DEFINITION A geometric sequece is a sequece such that for all, there is a costat r a such that a r The costat r is called the commo ratio 2 The recursive defiitio of a geometric sequece is: a a 2 r Whe writte i terms of a ad r, the terms of a geometric sequece are: a, a 2 = a r, a 3 = a r 2, a 4 = a r 3, Each term after the first is obtaied by multiplyig the previous term by r Therefore, each term is the product of a times r raised to a power that is oe less tha the umber of the term, that is: a a r 2 y Sice a sequece is a fuctio, we ca sketch the fuctio o the coordiate plae The geometric sequece, (2), (2) 2, (2) 3, (2) 4 or, 2, 4, 8, 6 ca be writte i fuctio otatio as {(,), (2, 2), (3, 4), (4, 8), (, 6)} Note that sice the domai is the set of positive itegers, the poits o the graph are distict poits that are ot coected by a curve May commo problems ca be characterized by a geometric sequece For example, if P dollars are ivested at a yearly rate of 4%, the the value of the ivestmet at the ed of each year forms a geometric sequece: O x
21 Year : EXAMPLE P 004P P( 004) P(04) Year 2: P(04) 004(P(04)) P(04)( 004) P(04)(04) P(04) 2 Year 3: P(04) 2 004(P(04) 2 ) P(04) 2 ( 004) P(04) 2 (04) P(04) 3 Year 4: P(04) 3 004(P(04) 3 ) P(04) 3 ( 004) P(04) 3 (04) P(04) 4 Year : P(04) 4 004(P(04) 4 ) P(04) 4 ( 004) P(04) 4 (04) P(04) The terms P(04), P(04) 2, P(04) 3, P(04) 4,, P(04) form a geometric sequece i which a P(04) ad r 04 The th term is a a r P(04)(04) Is the sequece 4, 2, 36, 08, 324, a geometric sequece? Solutio I the sequece, 4 3, 2 3, 36 3, 08 3, the ratio of ay term to the precedig term is a costat, 3, Therefore, 4, 2, 36, 08, 324, is a geometric sequece with a 4 ad r 3 EXAMPLE 2 What is the 0th term of the sequece 4, 2, 36, 08, 324,? Geometric Sequeces 267 Solutio The sequece 4, 2, 36, 08, 324, is a geometric sequece with a 4 ad r 3 Therefore, a a r Use a calculator for the computatio a 0 4(3) 9 ENTER: 4 3 ^ 9 ENTER DISPLAY: 4 * 3 9 > Aswer 78,732 Geometric Meas 4 I the proportio , we say that 20 is the mea proportioal betwee 4 ad 00 These three umbers form a geometric sequece 4, 20, 00 The mea proportioal, 20, is also called the geometric mea betwee 4 ad 00
22 268 Sequeces ad Series Betwee two umbers, there ca be ay umber of geometric meas For example, to write three geometric meas betwee 4 ad 00, we wat to form a geometric sequece 4, a 2, a 3, a 4, 00 I this sequece, a 4 ad a 00 There are two possible values of r, 2! ad! Therefore, there are two possible sequeces ad two possible sets of geometric meas Oe sequece is 4, 4!, 20, 20!, 00 with 4!, 20, ad 20! three geometric meas betwee 4 ad 00 The other sequece is 4, 24!, 20, 220!, 00 with 24!, 20, ad 220! three geometric meas betwee 4 ad 00 Note that for each sequece, the ratio of each term to the precedig term is a costat 4! 4! 20 4!!! 20! 20! 00 20!!!!! a a r 2 a a r r 4 2 r 4! 4 2! 4 r 4 r 6! 24! 4 2! 20 24! 2! 220! 20 2! ! 2!! 2 2!! 2 2! EXAMPLE 3 Fid four geometric meas betwee ad,2 Solutio We wat to fid the missig terms i the sequece, a 2, a 3, a 4, a,,2 Use the formula to determie the commo ratio r for a ad a 6,2 a a r 2 a 6 a r,2 r 243 r! 243! r r 3
23 Geometric Sequeces 269 The four geometric meas are (3), (3) 4, 4(3) 3, ad 3(3) 40 Aswer, 4, 3, ad 40 Exercises Writig About Mathematics Autum said that the aswer to Example 2 could have bee foud by eterig 4 3 ENTER o a calculator ad the eterig 3 ENTER eight times to display the sequece to the 9 terms after the first Do you thik that this is a easier way to fid the 0th term? Explai your aswer 2 Sierra said that 8, 8!2, 6, 6!2, 32 is a geometric sequece with three geometric meas, 8!2, 6, ad 6!2 Do you agree with Sierra? Justify your aswer Developig Skills I 3 4, determie whether each give sequece is geometric If it is geometric, fid r If it is ot geometric, explai why it is ot 3 4, 8, 6, 32, 64, 4,, 2, 2, 62, 3, 6, 9, 2, 6 2,2,8,32, 7, 23, 9, 227, 8, , 2, 4, 3, 9, 3, 9, 27, 3 0 2,2, 2,, 2,, 20, 00, 2,000, 0,000, 2, 0, 00, 000, 0000, 3 00, 20, 02, 204, 4 a, a 2, a 3, a 4, I 26, write the first five terms of each geometric sequece a, r 6 6 a 40, r 2 7 a 2, r 3 8 a 4, r 22 9 a, r!2 20 a 0, a a 2, a a 00, a 3 23 a, a a, a a, a a 8, a 27 What is the 0th term of the geometric sequece 02, 0,,? 28 What is the 9th term of the geometric sequece 2, 2,,? 29 I a geometric sequece, a ad a 6 Fid a 9 30 The first term of a geometric sequece is ad the 4th term is 27 What is the 8th term?
24 270 Sequeces ad Series 3 I a geometric sequece, a 2 ad a 3 6 Fid a 6 32 I a geometric sequece, a 3 ad a 7 9 Fid a 33 Fid two geometric meas betwee 6 ad Fid three geometric meas betwee 3 ad Fid three geometric meas betwee 8 ad 2,92 Applyig Skills 36 If $,000 was ivested at 6% aual iterest at the begiig of 200, list the geometric sequece that is the value of the ivestmet at the begiig of each year from 200 to Al ivested $3,000 i a certificate of deposit that pays % iterest per year What is the value of the ivestmet at the ed of each of the first four years? 38 I a small tow, a cesus is take at the begiig of each year The cesus showed that there were,000 people livig i the tow at the begiig of 200 ad that the populatio decreased by 2% each year for the ext seve years List the geometric sequece that gives the populatio of the tow from 200 to 2008 (A decrease of 2% meas that the populatio chaged each year by a factor of 098) Write your aswer to the earest iteger 39 It is estimated that the deer populatio i a park was icreasig by 0% each year If there were 0 deer i the park at the ed of the first year i which a study was made, what is the estimated deer populatio for each of the ext five years? Write your aswer to the earest iteger 40 A car that cost $20,000 depreciated by 20% each year Fid the value of the car at the ed of each of the first four years (A depreciatio of 20% meas that the value of the car each year was 080 times the value the previous year) 4 A maufacturig compay purchases a machie for $0,000 Each year the compay estimates the depreciatio to be % What will be the estimated value of the machie after each of the first six years? 66 GEOMETRIC SERIES DEFINITION A geometric series is the idicated sum of the terms of a geometric sequece For example, 3, 2, 48, 92, 768, 3,072 is a geometric sequece with r 4The idicated sum of this sequece, ,072, is a geometric series
25 Geometric Series 27 I geeral, if a, a r, a r 2, a r 3,,a r 2 is a geometric sequece with terms, the a a a a r a r 2 a r 3 c a r 2 i r i2 i is a geometric series Let the sum of these six terms be S 6 Now multiply S 6 by the egative of the commo ratio, 24 This will result i a series i which each term but the last is the opposite of a term i S 6 S ,072 24SS , ,288 23S ,288 S , ,096 4,09 Add the sums Divide by 23 Simplify Thus, S 6 4,09 The patter of a geometric series allows us to fid a formula for the sum of the series To S, add 2rS : S a a r a r 2 a r 3 c a r 2 2rS 2 a r 2 a r 2 2 a r 3 2 c 2 a r 2 2 a r S 2 rs a 2 a r S ( 2 r) a ( 2 r ) S a ( 2 r ) 2 r EXAMPLE Fid the sum of the first 0 terms of the geometric series 2 c 2 Solutio For the series 2 c, a 2 ad r S a ( 2 r ) 2 r S 0 2 A 2 A 2 B 0 B A 2,024 B 2 2 A,023,024 B 2 2 A,023,024 B 3 2, Aswer S 0,023 26
26 272 Sequeces ad Series EXAMPLE 2 Fid the sum of five terms of the geometric series whose first term is 2 ad whose fifth term is 62 Solutio Use a a r 2 to fid r 62 2r 4 8 r 4! 4 8 " 4 r 4 63 r S a ( 2 r ) 2 r S 2( 2 3 ) 2( 2 243) (2242) S 2( 2 (23) ) 2( 243) 2 (23) 4 2(244) 4 22 Aswer S 242 or S 22 Exercises Writig About Mathematics Casey said that the formula for the sum of a geometric series could be writte as S a 2 a r 2 r Do you agree with Casey? Justify your aswer 2 Sherri said that Example could have bee solved by simply addig the te terms of the series o a calculator Do you thik that this would have bee a simpler way of fidig the sum? Explai why or why ot Developig Skills I 3 4, fid the sum of terms of each geometric series 3 a, r 2, 2 4 a 4, r 3, a 2 6, r 4, 6 a 0, r 0, 6 7 a 3 04, r 2, 2 8 a, r 3, c a, c a, c a, c a, 7 3 a 4, a 324, 9 4 a, a 8 28, 0 I 22: a Write each sum as a series b Fid the sum of each series 3 a 3(2) 6 a a 0 A 2 B 8 26 a 26(4) A 3 B
27 Ifiite Series a (22) 20 a a 00 A 2 B k a k 23 Fid the sum of the first six terms of the series!3 3 3!3 24 Fid the sum of the first eight terms of a series whose first term is ad whose eighth term is 62 Applyig Skills 2 A group of studets are participatig i a math cotest Studets receive poit for their first correct aswer, 2 poits for their secod correct aswer, 4 poits for their third correct aswer, ad so forth What is the score of a studet who aswers 0 questios correctly? 26 Heidi deposited $400 at the begiig of each year for six years i a accout that paid % iterest At the ed of the sixth year, her first deposit had eared iterest for six years ad was worth 400(0) 6 dollars, her secod deposit had eared iterest for five years ad was worth 400(0) dollars, her third deposit had eared iterest for four years ad was worth 400(0) 4 dollars This patter cotiues a What is the value of Heidi s sixth deposit at the ed of the sixth year? Express your aswer as a product ad as a dollar value b Do the values of these deposits after six years form a geometric sequece? Justify your aswer c What is the total value of Heidi s six deposits at the ed of the sixth year? 27 A ball is throw upward so that it reaches a height of 9 feet ad the falls to the groud Whe it hits the groud, it bouces to 3 of its previous height If the ball cotiues i this way, boucig each time to 3 of its previous height util it comes to rest whe it hits the groud for the fifth time, fid the total distace the ball has traveled, startig from its highest poit 28 If you start a job for which you are paid $ the first day, $2 the secod day, $4 the third day, ad so o, how may days will it take you to become a millioaire? A 2 3 B k 28 A 2 3 B k 67 INFINITE SERIES Recall that a ifiite series is a series that cotiues without ed That is, for a series S a a 2 c a, we say that approaches ifiity We ca also ` a write a However, while there may be a ifiite umber of terms, a series ca behave i oly oe of three ways The series may icrease without limit, decrease without limit, or approach a limit
28 274 Sequeces ad Series CASE The series icreases without limit Cosider a arithmetic series S 2 (2a ( 2 )d) If a ad d 2: S S 2 S2() ( 2 ) A 8 2 B T 6 2 S T S T As the value of approaches ifiity, icreases without limit This series has o limit O We ca see this by graphig the fuctio S 4 for positive iteger values CASE 2 The series decreases without limit For a arithmetic series S 2 (2a ( 2 )d), if a ad d 2: S 2 f2() ( 2 )(2)g 2 f2 2 g 2 f3 2 g For this series, as the value of approaches 3 2 ifiity, or 2 2 f3 2 g 2 decreases without limit This series also has o limit We ca see this by graphig the fuctio S 2 for positive iteger values 2 O S CASE 3 The series approaches a limit a For a geometric series S 2 a r 2 r, if a ad r 2: 2 A 2 A S 2 B 2 B A 2 B 2 2 As approaches ifiity, A 2 B 2 A 2 B 2 A 2 B 2 approaches 0 Therefore, S 2 2 approaches S as approaches ifiity S 2 0 O
29 This series has a limit The diagram o the bottom of page 274 shows how the sum of terms approaches 2 as the umber of terms,, icreases I geeral, A ifiite arithmetic series has o limit A ifiite geometric series has o limit whe r + A ifiite geometric series has a fiite limit whe r * Whe r,, r approaches 0 Therefore: a S 2 a r a approaches 2 r 2 r as approaches ifiity Ifiite Series 27 EXAMPLE ` Fid a A 3 B Solutio a This is a ifiite geometric series with a 3 ad r 3 a Usig the formula 2 a r 2 r, ` A 3 B a A 3 B 2 3 As approaches ifiity, ` a ` A 3 B A 3 B A 3 B 3 A 3 B 2 A 3 B 3 c approaches 0 Therefore, Aswer Series Cotaiig Factorials Series that are either arithmetic or geometric must be cosidered idividually Ofte the limit of a series ca be foud to be fiite by comparig the terms of the series to the terms of aother series with a kow limit Oe such series is a k! k Recall factorials from previous courses We write factorial as follows: DEFINITION! ( 2 )( 2 2)( 2 3) c (3)(2)()
30 276 Sequeces ad Series Cosider the series a k!! 2! 3! c! This series ca be show to have a limit by comparig its terms to the terms of the geometric series with a ad r 2 Each term, a ( ), of this geometric series is 2 2 k Term of Term of Series Compariso Geometric Series to Be Tested of Terms! ! ! ! ! We ca coclude that! 2! 3! c!! ( 2 )( 2 2) c (2)() #, Therefore: Previously, we foud that for the geometric series with a ad r 2, the series approaches or 2 as approaches ifiity Therefore, a!, 2 Thus, 2 2 this series is bouded above by 2 To fid a lower boud, otice that the series! 2! 3! c! is the sum of positive terms Therefore, for 3,! 2!,! 2! 3! c! 3 or,! 2! 3! c! 2 Puttig it all together, what we have show is that for the series a! : 3 *! 2! 3! c 2! * 2 ` a 3 or 2 *! * c 2 2 ` `
31 Ifiite Series 277 The Number e Let us add to both sides of the iequality derived i the previous sectio We fid that as approaches ifiity, 3 2,! 2! 3! c!, 2 or 2,! 2! 3! c!, 3 The ifiite series! 2! 3! c! is equal to a irratioal umber that is greater tha 2 ad less tha 3 We call this umber e Eighteethcetury mathematicias computed this umber to may decimal places ad assiged to it the symbol e just as earlier mathematicias computed to may decimal places the ratio of the legth of the circumferece of a circle to the legth of the diameter ad assiged the symbol p to this ratio Therefore, we say: `! 2! 3! c a! ( 2 )! c e 0 This umber has a importat role i may differet braches of mathematics A calculator will give the value of e to ie decimal places ENTER: 2d e x ENTER DISPLAY: e ( > EXAMPLE 2 Fid, to the earest hudredth, the value of e 2 Solutio Evaluate the expressio o a calculator ENTER: 2d e x 2 ENTER DISPLAY: +e (2 > Aswer To the earest hudredth, e 2 839
32 278 Sequeces ad Series Exercises Writig About Mathematics Show that if the first term of a ifiite geometric series is ad the commo ratio is c, the c the sum is c 2 2 Cody said that sice the calculator gives the value of e as , the value of e ca be writte as 27828, a repeatig decimal ad therefore a ratioal umber Do you agree with Cody? Explai why or why ot Developig Skills I 3 0: a Write each series i sigma otatio b Determie whether each sum icreases without limit, decreases without limit, or approaches a fiite limit If the series has a fiite limit, fid that limit 3 3 9??? ??? ??? 6 2??? ??? ??? 9 2! 3! c ( )! c ??? I 6, a ifiitely repeatig decimal is a ifiite geometric series Fid the ratioal umber represeted by each of the followig ifiitely repeatig decimals ??? ??? ??? The sum of the ifiite series 2 c is 2 Fid values of such that a,
33 Chapter Summary 279 CHAPTER SUMMARY A sequece is a set of umbers writte i a give order Each term of a sequece is associated with the positive iteger that specifies its positio i the ordered set A fiite sequece is a fuctio whose domai is the set of itegers {, 2, 3,, }A ifiite sequece is a fuctio whose domai is the set of positive itegers The terms of a sequece are ofte desigated as a, a 2, a 3,The formula that allows ay term of a sequece except the first to be computed from the previous term is called a recursive defiitio A arithmetic sequece is a sequece such that for all, there is a costat d such that a + 2 a d For a arithmetic sequece: a a ( 2 )d a 2 d A geometric sequece is a sequece such that for all, there is a costat r a such that a r For a geometric sequece: a a r 2 a 2 r A series is the idicated sum of the terms of a sequece The Greek letter S is used to idicate a sum defied for a set of cosecutive iteger If S represets the th partial sum, the sum of the first terms of a sequece, the S a For a arithmetic series: For a geometric series: a k k a a 2 a 3??? a S 2 (a a ) 2 (2a ( 2 )d) S a ( 2 r ) 2 r For a geometric series, if r, ad approaches ifiity: S a 2 r As approaches ifiity: or ` a a r 2 a 2 r `! 2! 3! c a! ( 2 )! e 0 The umber e is a irratioal umber
34 280 Sequeces ad Series VOCABULARY 6 Sequece Fiite sequece Ifiite sequece Recursive defiitio 62 Commo differece Arithmetic sequece Arithmetic meas 63 Series S Sigma otatio Fiite series Ifiite series ` 64 Arithmetic series S th partial sum 6 Geometric sequece Commo ratio Geometric mea 66 Geometric series 67 factorial e REVIEW EXERCISES I 6: a Write a recursive formula for a b Is the sequece arithmetic, geometric, or either? c If the sequece is arithmetic or geometric, write a explicit formula for a d Fid a 0,, 9, 3, 2 3,, 3, 9, 3 2,, 0, 9, 4, 3, 6, 0,, i,3i,7i,i,3i, 6 2, 26, 8, 24, 62, I 7 2, write each series i sigma otatio c I a arithmetic sequece, a 6 ad d Write the first five terms 4 I a arithmetic sequece, a 6 ad d Fid a 30 I a arithmetic sequece, a ad a 4 23 Fid a 2 6 I a arithmetic sequece, a 3 0 ad a 0 70 Fid a 7 I a geometric sequece, a 2 ad a 2 Write the first five terms 8 I a geometric sequece, a 2 ad a 2 Fid a 0 9 I a geometric sequece, a 2 ad a 4 28 Fid a 6 20 Write a recursive formula for the sequece 2, 20, 30, 42, 6, 70, 90, 0,
35 Review Exercises 28 2 Write the first five terms of a sequece if a ad a 4a 3 22 Fid six arithmetic meas betwee ad Fid three geometric meas betwee ad I a arithmetic sequece, a, d 8 If a 89, fid 2 I a geometric sequece, a ad a 4 Fid r 4 26 Write a 3 as the sum of terms ad fid the sum 6 27 Write a (2 2 3k) as the sum of terms ad fid the sum k0 28 To the earest hudredth, fid the value of 3e 3 29 a Write, i sigma otatio, the series ??? b Determie whether the series give i part a icreases without limit, decreases without limit, or approaches a fiite limit If the series has a fiite limit, fid that limit 30 A grocer makes a display of caed tomatoes that are o sale There are 24 cas i the first (bottom) layer ad, after the first, each layer cotais three fewer cas tha i the layer below There are three cas i the top layer a How may layers of cas are i the display? b If the grocer sold all of the cas i the display, how may cas of tomatoes did he sell? 3 A retail store pays cashiers $20,000 a year for the first year of employmet ad icreases the salary by $00 each year a What is the salary of a cashier i the teth year of employmet? b What is the total amout that a cashier ears i his or her first 0 years? 32 Be started a job that paid $40,000 a year Each year after the first, his salary was icreased by 4% a What was Be s salary i his eighth year of employmet? b What is the total amout that Be eared i eight years? 33 The umber of hadshakes that are exchaged if every perso i a room shakes hads oce with every other perso ca be writte as a sequece Let a 2 be the umber of hadshakes exchaged whe there are 2 persos i the room If oe more persos eters the room so that there are persos i the room, that perso shakes hads with 2 persos, creatig 2 additioal hadshakes a Write a recursive formula for a b If a 0, write the first 0 terms of the sequece c Write a formula for a i terms of
36 282 Sequeces ad Series Exploratio The graphig calculator ca be used to graph recursive sequeces ad examie the covergece, divergece, or oscillatio of the sequece A sequece coverges if as goes to ifiity, the sequece approaches a limit A sequece diverges if as goes to ifiity, the sequece icreases or decreases without limit A sequece oscillates if as goes to ifiity, the sequece either coverges or diverges For example, to verify that the geometric sequece a a 2 A 2 B with a coverges to 0: STEP Set the calculator to sequece graphig mode Press MODE ad select Seq, the last optio i the fourth row Normal Sci Eg Float Radia Degree Fuc Par Pol Seq Coected Dot Sequetial Simul Real a+bi re u i Full Horiz GT > STEP 2 Eter the expressio u( 2 ) A 2 B a 2 A 2 B as ito u() The fuctio ame u is foud above the 7 key ENTER: Y 2d u ( X,T,, ) 2 Plot Plot 2 Plot3 Mi= u() = u()*/2 u(mi) = {} v()= v(mi)= w()= STEP 3 To set the iitial value of a, set u(mi) ENTER: (To set more tha oe iitial value, for example a ad a 2 6, set u(mi) {, 6}) STEP 4 Graph the sequece up to 2 Press WINDOW Max 2 2 to set WINDOW Mi= Max=2 PlotStart= PlotStep= Xmi=  0 Xmax=0 Xscl= >
37 Cumulative Review 283 STEP Fially, press ZOOM to graph the sequece You ca use TRACE to explore the values of the sequece We ca see that the sequece approaches the value of 0 very quickly 0 I 4: a Graph each sequece up to the first 7 terms b Use the graph to make a cojecture as to whether the sequece coverges, diverges, or oscillates a, a 2, a = a 2 a (Fiboacci sequece) 2 a, a a 2 3 a, a 00 2 a a, a 09a 2 00(07) 0 CUMULATIVE REVIEW CHAPTERS 6 Part I Aswer all questios i this part Each correct aswer will receive 2 credits No partial credit will be allowed If the domai is the set of itegers, the the solutio set of 22 # x 3, 7 is () {22, 2, 0,, 2, 3, 4,, 6, 7} (2) {22, 2, 0,, 2, 3, 4,, 6} (3) {2, 24, 23, 22, 2, 0,, 2, 3, 4} (4) {2, 24, 23, 22, 2, 0,, 2, 3} 2 The expressio (a 3) 2 2 (a 3) is equal to () a 2 2 a 6 (3) a 2 a 2 (2) a 2 a 6 (4) a 3 x 3 I simplest form, 3 x x is equal to x 2 6 x 6 () 24 (3) 24 x 2 8 x(x 2 2)2 (2) 24 (4) 24
38 284 Sequeces ad Series 4 The sum of A3 2!B A2 2!4B is () 2 4! (2) 2 3! (3) 2!2 (4)!2 The fractio is equal to 2!2 ()!2 (2) 2!2 (3) 2 2!2 (4) 2!2 6 The quadratic equatio 3x 2 2 7x 3 has roots that are () real, ratioal, ad equal (2) real, ratioal, ad uequal (3) real irratioal ad uequal (4) imagiary 7 A fuctio that is oetooe is () y 2x (3) y x 4 x (2) y x 2 (4) y x 8 The fuctio y = x 2 is traslated 2 uits up ad 3 uits to the left The equatio of the ew fuctio is () y (x 2 3) 2 2 (3) y (x 3) 2 2 (2) y (x 2 3) (4) y (x 3) Which of the followig is a real umber? () i (2) i 2 (3) i 3 (4) 2i 0 The sum of the roots of a quadratic equatio is 24 ad the product of the roots is The equatio could be () x 2 2 4x 0 (3) x 2 2 4x 2 0 (2) x 2 4x 2 0 (4) x 2 4x 0 Part II Aswer all questios i this part Each correct aswer will receive 2 credits Clearly idicated the ecessary steps, icludig appropriate formula substitutios, diagrams, graphs, charts, etc For all questios i this part, a correct umerical aswer with o work will receive oly credit What umber must be added to each side of the equatio x 2 3x 0 i order to make the left member the perfect square of a biomial? 2 Solve for x ad check: 7 2!x 2 4
39 Cumulative Review 28 Part III Aswer all questios i this part Each correct aswer will receive 4 credits Clearly idicated the ecessary steps, icludig appropriate formula substitutios, diagrams, graphs, charts, etc For all questios i this part, a correct umerical aswer with o work will receive oly credit (2 i) 3 Express the fractio 2 i i a bi form 4 Graph the solutio set of the iequality 2x 2 2 x Part IV Aswer all questios i this part Each correct aswer will receive 6 credits Clearly idicated the ecessary steps, icludig appropriate formula substitutios, diagrams, graphs, charts, etc For all questios i this part, a correct umerical aswer with o work will receive oly credit a Write the fuctio that is the iverse fuctio of y 3x b Sketch the graph of the fuctio y 3x ad of its iverse c What are the coordiates of the poit of itersectio of the fuctio ad its iverse? 6 a Write a recursive defiitio for the geometric sequece of five terms i which a 3 ad a 30,000 b Write the sum of the sequece i part a i sigma otatio c Fid the sum from part b
Basic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (1226) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationListing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2
74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is
More informationCHAPTER 11 Financial mathematics
CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula
More information23 The Remainder and Factor Theorems
 The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationTHE ARITHMETIC OF INTEGERS.  multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS  multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible sixfigure salaries i whole dollar amouts are there that cotai at least
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a page formula sheet. Please tur over Mathematics/P DoE/November
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationFM4 CREDIT AND BORROWING
FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer
More informationMATH 083 Final Exam Review
MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period
More informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationFOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRECALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationSolving equations. Pretest. Warmup
Solvig equatios 8 Pretest Warmup We ca thik of a algebraic equatio as beig like a set of scales. The two sides of the equatio are equal, so the scales are balaced. If we add somethig to oe side of the
More informationQuestion 2: How is a loan amortized?
Questio 2: How is a loa amortized? Decreasig auities may be used i auto or home loas. I these types of loas, some amout of moey is borrowed. Fixed paymets are made to pay off the loa as well as ay accrued
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability betwee group variability withi group variability total variability Fratio Computatio sums of squares (betwee/withi/total degrees of freedom (betwee/withi/total mea square (betwee/withi
More informationTime Value of Money. First some technical stuff. HP10B II users
Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationLearning objectives. Duc K. Nguyen  Corporate Finance 21/10/2014
1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the timevalue
More information3. If x and y are real numbers, what is the simplified radical form
lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a otforprofit membership associatio whose missio is to coect studets to college success
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More information3. Greatest Common Divisor  Least Common Multiple
3 Greatest Commo Divisor  Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationEngineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More informationSequences and Series Using the TI89 Calculator
RIT Calculator Site Sequeces ad Series Usig the TI89 Calculator Norecursively Defied Sequeces A orecursively defied sequece is oe i which the formula for the terms of the sequece is give explicitly. For
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX9850GB PLUS to efficietly compute values associated with preset value auities.
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More information2 Time Value of Money
2 Time Value of Moey BASIC CONCEPTS AND FORMULAE 1. Time Value of Moey It meas moey has time value. A rupee today is more valuable tha a rupee a year hece. We use rate of iterest to express the time value
More informationFI A CIAL MATHEMATICS
CHAPTER 7 FI A CIAL MATHEMATICS Page Cotets 7.1 Compoud Value 117 7.2 Compoud Value of a Auity 118 7.3 Sikig Fuds 119 7.4 Preset Value 122 7.5 Preset Value of a Auity 122 7.6 Term Loas ad Amortizatio 123
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationCDs Bought at a Bank verses CD s Bought from a Brokerage. Floyd Vest
CDs Bought at a Bak verses CD s Bought from a Brokerage Floyd Vest CDs bought at a bak. CD stads for Certificate of Deposit with the CD origiatig i a FDIC isured bak so that the CD is isured by the Uited
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationMATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12
Mathematics/P1 1 Jue 014 Commo Test MATHEMATICS P1 COMMON TEST JUNE 014 NATIONAL SENIOR CERTIFICATE GRADE 1 Marks: 15 Time: ½ hours N.B: This questio paper cosists of 7 pages ad 1 iformatio sheet. Please
More informationEGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES
EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND OBTAINED WITH ENGEL SERIES ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Abstract. The aciet Egyptias epressed ratioal
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea  add up all
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationBENEFITCOST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEITCST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal  Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationDescriptive Statistics
Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measuretheoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real oegative umber R, called the radius
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a stepbystep procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationMultiple Representations for Pattern Exploration with the Graphing Calculator and Manipulatives
Douglas A. Lapp Multiple Represetatios for Patter Exploratio with the Graphig Calculator ad Maipulatives To teach mathematics as a coected system of cocepts, we must have a shift i emphasis from a curriculum
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationOverview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals
Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More information