Bachelor in Economics (S.E): ManajemenCourse : Matematika Ekonomi (1508ME12)online.uwin.ac.idSession Topic : Dasar2 Deret Hitung & Ukurserta AplikasinyaCourse: Matematika EkonomiBy Handri Santoso, Ph.DUWIN eLearning ProgramPowered by HarukaEdu.com - 1508ME12-S.3Content Part 1 Sequences & Series Part 2 Arithmetic Sequences (Deret Hitung) Part 3 Geometric Sequences (Deret Ukur) Part 4 Infinite Sequences (Deret Ukur Tak Terbatas)Part1: Sequences & SeriesPowered by HarukaEdu.com - 1508ME12-S.5Sequences & Series: Definition1. Sequence Defn: A set of numbers that follow a general rule (n-th term formula) Examples: a. 1, 8, 15, 22, 29, b. -8, 2, -1/2, 1/8, 2. Series Defn: The sum of the terms in a sequence of numbers Examples:a. 1 + 8 + 15 + 22 + 29 + b. -8 + 2 1/2 + 1/8 Powered by HarukaEdu.com - 1508ME12-S.6Sequences & Series: Finite vs Infinite1. A Finite series has an endExamples: a. -9 3 + 3 + 9 + 15b. 2 + 4 + 8 + + 10242. An Infinite series continues indefinitelyExamples:16 + 12 + 9 + Part2: Arithmetic Sequences (Deret Hitung)Powered by HarukaEdu.com - 1508ME12-S.8Arithmetic Sequence: DefinitionA sequence,a1, a2, a3,.., an, Is called an arithmetic sequence, or arithmetic progression, If there exists a constant d, called the common difference, such thatan an-1 = dThat is,an = an-1 + dfor every n > 1nth Term FormulasIf {an} is an arithmetic sequence with common difference d, then1) a2 = a1 + d 2) a3 = a2 + d= a1 + 2d 3) a4 = a3 + d= a1 + 3d The nth Term of an Arithmetic Sequencean = a1 + (n-1)dfor every n > 1Powered by HarukaEdu.com - 1508ME12-S.9Arithmetic Sequence: ExampleFind the next four terms of,1) 9, -2, 5, Arithmetic Sequence-2 -9 = 5 -2 = 7 7 is referred to as the common difference (d) Common Difference (d) what we ADD to get next term Next four terms12, 19, 26, 332) 0, 7, 14, Arithmetic Sequence, d = 721, 28, 35, 423) x, 2x, 3x, Arithmetic Sequence, d = x4x, 5x, 6x, 7x4) 5k, -k, -7k, Arithmetic Sequence, d = -6k-13k, -19k, -25k, -32kPowered by HarukaEdu.com - 1508ME12-S.10Arithmetic Sequence: Sum FormulasSum Formulas for Finite Arithmetic Series If a1, a2, a3,.., anis a finite arithmetic sequence, then the corresponding series a1+ a2+ a3+..+ anis called an arithmetic series. We will derive 2 simple & very useful formulas for the sum of an arithmetic series. Let d be the common difference of the arithmetic sequence a1, a2, a3,.., an& let Sndenote the sum of the series a1+ a2+ a3+..+ anThenSn= a1+ (a1 + d ) + + [a1 + (n 2) d] + [a1 + (n 1) d] Reversing the order of the sum, we obtainSn= [a1 + (n 1) d] + [a1 + (n 2) d] + + (a1 + d ) + a1 Powered by HarukaEdu.com - 1508ME12-S.11Arithmetic Sequence: Sum Formulas (Cont.) Adding the left sides of these 2 equations & corresponding elements of the right sides, we see that2Sn = [2a1+ (n 1)d] + [2a1+ (n 1)d] + + [2a1+ (n 1)d] = n[2a1+ (n 1)d] Sum ofan Aritmetic Series,1) First FormBy replacing a1+ (n 1)d with an, we obtain a second useful formula for the sum2) Second FormPowered by HarukaEdu.com - 1508ME12-S.12Arithmetic Sequence: Exercise1.) Given an arithmetic sequence with a15= 38 & d = -3,Find a1 x3815NA-3a1 First terman nth termn Number of termsSnSum of n termsd Common differencean = a1 + (n-1)d38 = x + (15-1)(-3)x = 802.) Find S63of -19, -13. -7, -19??63x6a1 First terman nth termn Number of termsSnSum of n termsd Common difference an= a1 + (n-1)d?? = -19 + (63-1)(6)?? = 353 Sn= (a1 + an)S63= (-19 + 353)S63= 10521n2632Powered by HarukaEdu.com - 1508ME12-S.13Arithmetic Sequence: Exercise (Cont.)3.) Find a16 , if a1= 1.5 & d = 0.51.5x16NA0.5a1 First terman nth termn Number of termsSnSum of n termsd Common differencean = a1 + (n-1)da16= 1.5 + (16-1)(0.5)a16= 94.) Find n, if an= 633, a1= 9 & d = 249633xNA24a1 First terman nth termn Number of termsSnSum of n termsd Common differencean = a1 + (n-1)d633 = 9 + (x-1)(24)633 = 9 + 24x - 24x = 27Powered by HarukaEdu.com - 1508ME12-S.14Arithmetic Sequence: Exercise (Cont.)5.) Find d , if a1= -6 & a29= 20-62029NAxa1 First terman nth termn Number of termsSnSum of n termsd Common differencean = a1 + (n-1)d20 = -6 + (29-1)x26 = 28xx = 136.) Find 2 arithmetic means between-4 & 5-4, __, ___, 5-454NAxa1 First terman nth termn Number of termsSnSum of n termsd Common differencean = a1 + (n-1)d5 = -4 + (4-1)(x)x = 3The 2 arithmetic means are -1 & 2, since -4, -1, 2, 5 forms an arithmetic sequence14Powered by HarukaEdu.com - 1508ME12-S.15Arithmetic Sequence: Exercise (Cont.)7.) Find 3 arithmetic means between1 & 41, __, ___, ___, 4145NAxa1 First terman nth termn Number of termsSnSum of n termsd Common differencean = a1 + (n-1)d4 = 1 + (5-1)(x)x = 3The 3 arithmetic means are 7/4, 10/4 & 13/4, since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence4Powered by HarukaEdu.com - 1508ME12-S.16Arithmetic Sequence: Exercise (Cont.)8.) Find n for the series in which a1= 5, d = 3 Sn= 4405yx4403a1 First terman nth termn Number of termsSnSum of n termsd Common differencean = a1 + (n-1)dy = 5 + (x-1)3Sn= (a1 + an)440 = (5 + y)440 = (5 + 5 + (x-1)3)440 = 880 = x(7+3x)0 = 3x2+ 7x -880Graph on positive windowX = 16n2n2x2x(7+3x)2Powered by HarukaEdu.com - 1508ME12-S.17Arithmetic Sequence: nth Partial SumThe sum of the first n terms of an infinite sequence is called the nth partial sum. Example: The nth Partial Sum1.) Find the 150thpartial sum ofthe arithmetic sequence, 5, 16, 27, 38, 49, a1= 5, d = 11 c = 5 11 = -6o an= 11n 6a150= 11(150) 6 = 1644o S150= (5 + 1644) = 75(1649) = 123,6751502Powered by HarukaEdu.com - 1508ME12-S.18Arithmetic Sequence: nth Partial Sum (Cont.)2.) An auditorium has 20 rows ofseats. There are,a) 20 seats in the first row, b) 21 seats in the second row, c) 22 seats in the third row, &d) so on. How many seats are there in all 20 rows? d = 1 c = 201 = 19o an = a1 + (n-1)da20= 20 + 19(1) = 39o S20= (20 + 39) = 10(59) = 590202Powered by HarukaEdu.com - 1508ME12-S.19Arithmetic Sequence: nth Partial Sum (Cont.)3.) A small business sells $10,000 worth ofsports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation. a1= 10,000 d = 7500 c = 10,0007500 = 2500o an = a1 + (n-1)da20= 10,000 + 19(7500) = 152,500o S20= (10,000 + 152,500) = 10(162,500) = 1,625,000So the total sales for the first 20 years is $1,625,000202Part3: Geometric Sequences (Deret Ukur)Powered by HarukaEdu.com - 1508ME12-S.21Geometric Sequence: DefinitionA sequence,a1, a2, a3,.., an, Is called an geometric sequence, or geometric progression, If there exists a nonzero constant r, called the common ratio, such thatThat is,an = ran-1for every n > 1

= Similarly, if {an} is a geometric sequence with common ratio rthen1) a2 = a1r2) a3 = a2r = a1r23) a4 = a3r = a1r3The nth Term of a Geometric Sequencean = a1r n-1for every n > 1Powered by HarukaEdu.com - 1508ME12-S.22Geometric Sequence: Sum FormulasSum Formulas for Finite Geometric Series If a1, a2, a3,.., anis a finite geometric sequence, then the corresponding series a1+ a2+ a3+..+ anis called a geometric series. As with arithmetic series, we can derive 2 simple & very useful formulas for the sum of a geometric series. Let r be the common ratio of the geometric sequence a1, a2, a3,.., an& let Sndenote the sum of the series a1+ a2+ a3+..+ anThenSn= a1+ a1r + a1r2+ a1r3+ + a1rn-2 + a1rn-1 Multiply both sides of this equation by rto obtainrSn= a1r + a1r2+ a1r3+ + a1rn-1 + a1rnPowered by HarukaEdu.com - 1508ME12-S.23r 1Geometric Sequence: Sum Formulas (Cont.) Now subtract the left side of the second equation from the left side of the first, & The right side of the second equation from the right side of the first to obtainSn rSn= a1 a1rnSn(1r)= a1 a1rnThus, solving for Sn, we obtain the following formula for the sum of a geometric series:Sum ofa Geometric Series,1) First Form

= a1rn Since an= a1rn-1or ran= a1rn, the sum formula also can be written in the following form:2) Second Form

=

r 1Powered by HarukaEdu.com - 1508ME12-S.24Geometric Sequence: Vocabulary & FiniteVocabulary ofSequences (Universal)a1 First terman nth termn Number of termsSnSum of n termsr Common ratio nth term of geometric sequencean = a1rn-1 Sum of n term of geometric sequence Finite Geometric SeriesWARNING: you must have a Geometric Series to use this formula

= [

(

)]

=

) Powered by HarukaEdu.com - 1508ME12-S.25Geometric Sequence: Exercise1.) Find the next 3 terms of2, 3, 9/2, ___, ___, ___3 2 vs. 9/2 3 not arithmetic2.) If a1= 1/2 r= 2/3Find a9 1/2x9NA2/3a1 First terman nth termn Number of termsSnSum of n termsr Common ratio32 = 9/23= 1.5geometric = 323 3 3 3 3 32 2 292, 3, , , ,2 9 9 92 2 2 2 2 2 92, 3, , ,27 81 2434 8 ,2 16 an = a1rn-1 9 11 2x2 3| || |=| |\ .\ .882x2 3=7823=1286561=Powered by HarukaEdu.com - 1508ME12-S.26Geometric Sequence: Exercise (Cont.)4.) Find a2 a4 if a1= -3 r= 2/3-3, __, ___, ___Since3.) Find 2 geometric means between-2 & 54-2, __, ___, 54-2544NAxa1 First terman nth termn Number of termsSnSum of n termsr Common ratioan = a1rn-1 54 = (-2)(x)4-1 -27 = x3-3 = xThe 2 geometric means are 6 & -18, since -2, 6, 18, 54 forms an geometric sequence = 23 3, 2,43 ,89 2 4 = 2 89= 109Powered by HarukaEdu.com - 1508ME12-S.27Geometric Sequence: Exercise (Cont.)6.) If a5= 322 r= - 2Find a2___, __, ___, ___, 3225.) Find a9 of 2, 2, 222x9NAa1 First terman nth termn Number of termsSnSum of n termsr Common ratioan = a1rn-1 x = 2(2)9-1 x = 2(2)8x = 162 = 22= 222= 2x322 5NA-2a1 First terman nth termn Number of termsSnSum of n termsr Common ratioan = a1rn-1 32 2 = x(-2)5-1 32 2 = x (-2)4 32 2 = 4x82 = xPowered by HarukaEdu.com - 1508ME12-S.28Geometric Sequence: Exercise (Cont.)7.) Insert one geometric mean between & 4 denotes trick question , ___, 4 43NAxa1 First terman nth termn Number of termsSnSum of n termsr Common ratioo an = a1rn-1 3 1144r =2r144=216 r = 4 r =1, 1, 441, 1, 44 Powered by HarukaEdu.com - 1508ME12-S.29Geometric Sequence: Exercise (Cont.)8.) Find S7of .. NA7xa1 First terman nth termn Number of termsSnSum of n termsr Common ratio12 +14 +18 + = 1412 = 1814 = 12

= [1

1 ] 1 =12 (12)7112 1=12 (12)7112=

Part4: Infinite Sequences (Deret Ukur Tak Terbatas)Powered by HarukaEdu.com - 1508ME12-S.31Geometric Sequence: Sum FormulaSum Formula for Infinite Geometric Series Consider a geometric series with a1=5 & r = . What happens to the sum Sn as n increases ? To answer this question, we first write the sum formula in the more convenient form

=

=

For a1=5 & r = ,

= 10 10 12

(1)Powered by HarukaEdu.com - 1508ME12-S.32Geometric Sequence: Sum Formula (Cont.)Thus, It appears that becomes smaller & smaller as n increases & the sum gets closer & closer to 10 In general, it is possible to show that, if |r|