07/11/2017 BARISAN DAN DERET KONSEP BARISAN DAN DERET 1

07/11/2017 SEQUENCE AND SERIES THE CONCEPT OF SEQUENCE AND SERIES 2

Menerapkan konsep barisan dan deret aritmatika 07/11/2017 Pola Barisan dan Deret Bilangan Kompetensi Dasar : Menerapkan konsep barisan dan deret aritmatika Indikator : Nilai suku ke- n suatu barisan aritmatika ditentukan menggunakan rumus Jumlah n suku suatu deret aritmatika ditentukan dengan menggunakan rumus Hal.: 3 Hal.: 3 BARISAN DAN DERET 3

Applying the concept of arithmetic sequence and series 07/11/2017 The Pattern of Sequence and Series Number Basic Competence: Applying the concept of arithmetic sequence and series Indicator : The value of n-th term in an arithmetic sequence is defined by formula The sum of n in term of arithmetic sequence is defined by formula Hal.: 4 Hal.: 4 BARISAN DAN DERET 4

Pola Barisan dan Deret Bilangan 07/11/2017 Pola Barisan dan Deret Bilangan Saat mengendarai motor, pernahkah kalian mengamati speedometer pada motor tersebut? Pada speedometer terdapat angka-angka 0,20, 40, 60, 80, 100, dan 120 yang menunjukkan kecepatan motor saat kalian mengendarainya. Angka-angka ini berurutan mulai dari yang terkecil ke yang terbesar dengan pola tertentu sehingga membentuk sebuah pola barisan Hal.: 5 Hal.: 5 BARISAN DAN DERET 5

The Pattern of Sequence and Series Number 07/11/2017 The Pattern of Sequence and Series Number When you ride a motor cycle, have you ever look at the speeedometer? In speedometer,there are numbers of 0,20, 40, 60, 80, 100, and 120 which show the speed of your motor cycle. These numbers are un order, starts from the smallest to the biggest with certain pattern, so that it forms a pattern of sequence Hal.: 6 Hal.: 6 BARISAN DAN DERET 6

Pola Barisan dan Deret Bilangan 07/11/2017 Pola Barisan dan Deret Bilangan Bayangkan anda seorang penumpang taksi. Dia harus membayar biaya buka pintu Rp 15.000 dan argo Rp 2.500 /km. Buka pintu 1 km 2 km 3 km 4 km 15.000 17.500 20.000 22.500 ……. Hal.: 7 Hal.: 7 BARISAN DAN DERET 7

The Pattern of Sequence and Series Number 07/11/2017 The Pattern of Sequence and Series Number Imagine that you are a taxi passenger. You have to pay the starting fee Rp 15.000 and it charge Rp 2.500 /km. Starting fee 1 km 2 km 3 km 4 km 15.000 17.500 20.000 22.500 ……. Hal.: 8 BARISAN DAN DERET 8

NOTASI SIGMA Konsep Notasi Sigma 07/11/2017 NOTASI SIGMA Konsep Notasi Sigma Perhatikan jumlah 6 bilangan ganjil pertama berikut: 1 + 3 + 5 + 7 + 9 + 11 ……….. (1) Pada bentuk (1) Suku ke-1 = 1 = 2.1 – 1 Suku ke-2 = 3 = 2.2 – 1 Suku ke-3 = 5 = 2.3 – 1 Suku ke-4 = 7 = 2.4 – 1 Suku ke-5 = 9 = 2.5 – 1 Suku ke-6 = 11 = 2.6 – 1 Secara umum suku ke-k pada (1) dapat dinyatakan dalam bentuk 2k – 1, k  { 1, 2, 3, 4, 5, 6 } Hal.: 9 BARISAN DAN DERET 9

The Concept of Sigma Notation 07/11/2017 SIGMA NOTATION The Concept of Sigma Notation Look at the sum of the first sixth odd number below: 1 + 3 + 5 + 7 + 9 + 11 ……….. (1) In the form(1) The 1st term = 1 = 2.1 – 1 The 2nd term= 3 = 2.2 – 1 The 3rd term = 5 = 2.3 – 1 The 4th term = 7 = 2.4 – 1 The 5th term = 9 = 2.5 – 1 The 6th term = 11 = 2.6 – 1 Generally, the k-th term in (1) can be stated in the form of 2k – 1, k  { 1, 2, 3, 4, 5, 6 } Hal.: 10 BARISAN DAN DERET 10

NOTASI SIGMA Dengan notasi sigma bentuk penjumlahan (1) dapat 07/11/2017 NOTASI SIGMA Dengan notasi sigma bentuk penjumlahan (1) dapat ditulis : Hal.: 11 BARISAN DAN DERET 11

07/11/2017 SIGMA NOTATION In Sigma notation, the addition form (1) can be written as: Hal.: 12 BARISAN DAN DERET 12

1 disebut batas bawah dan 6 disebut batas atas, k dinamakan indeks 07/11/2017 NOTASI SIGMA Bentuk dibaca “sigma 2k – 1 dari k =1 sampai dengan 6” atau “jumlah 2k – 1 untuk k = 1 sd k = 6” 1 disebut batas bawah dan 6 disebut batas atas, k dinamakan indeks (ada yang menyebut variabel) Hal.: 13 BARISAN DAN DERET 13

1 is called lower limit and 6 is called upper limit, 07/11/2017 SIGMA NOTATION In the form of It is read “sigma 2k – 1 from k =1 to 6” or “the sum of 2k – 1 for k = 1 sd k = 6” 1 is called lower limit and 6 is called upper limit, k is called index (some people called it variable) Hal.: 14 BARISAN DAN DERET 14

07/11/2017 NOTASI SIGMA Secara umum Hal.: 15 BARISAN DAN DERET 15

07/11/2017 SIGMA NOTATION Generally Hal.: 16 BARISAN DAN DERET 16

Nyatakan dalam bentuk sigma 07/11/2017 NOTASI SIGMA Contoh: Hitung nilai dari: Nyatakan dalam bentuk sigma 1. a + a2b + a3b2 + a4b3 + … + a10b9 Hal.: 17 BARISAN DAN DERET 17

Example: Define the value of Stated into sigma form 07/11/2017 SIGMA NOTATION Example: Define the value of Stated into sigma form 1. a + a2b + a3b2 + a4b3 + … + a10b9 Hal.: 18 BARISAN DAN DERET 18

07/11/2017 NOTASI SIGMA 2. (a + b)n = Hal.: 19 BARISAN DAN DERET 19

07/11/2017 SIGMA NOTATION 2. (a + b)n = Hal.: 20 BARISAN DAN DERET 20

Sifat-sifat Notasi Sigma : 07/11/2017 NOTASI SIGMA Sifat-sifat Notasi Sigma : , Untuk setiap bilangan bulat a, b dan n Hal.: 21 BARISAN DAN DERET 21

The properties of sigma notation : 07/11/2017 SIGMA NOTATION The properties of sigma notation : , For every integer a, b and n Hal.: 22 BARISAN DAN DERET 22

NOTASI SIGMA Contoh1: Tunjukkan bahwa Jawab : Hal.: 23 07/11/2017 NOTASI SIGMA Contoh1: Tunjukkan bahwa Jawab : Hal.: 23 BARISAN DAN DERET 23

SIGMA NOTATION Example 1: Show that Answer : Hal.: 24 07/11/2017 SIGMA NOTATION Example 1: Show that Answer : Hal.: 24 BARISAN DAN DERET 24

NOTASI SIGMA Contoh 2 : Hitung nilai dari Jawab: 07/11/2017 NOTASI SIGMA Contoh 2 : Hitung nilai dari Jawab: = 6 (12 +22 + 32 + 42 + 52 + 62) = 6 (1 + 4 + 9 + 16 + 25 + 36) = 6.91 = 546 Hal.: 25 BARISAN DAN DERET 25

SIGMA NOTATION Example 2 : Define the value of Answer: 07/11/2017 SIGMA NOTATION Example 2 : Define the value of Answer: = 6 (12 +22 + 32 + 42 + 52 + 62) = 6 (1 + 4 + 9 + 16 + 25 + 36) = 6.91 = 546 Hal.: 26 BARISAN DAN DERET 26

BARISAN DAN DERET ARITMATIKA 07/11/2017 BARISAN DAN DERET ARITMATIKA Bilangan-bilangan berurutan seperti pada speedometer memiliki selisih yang sama untuk setiap dua suku berurutannya sehingga membentuk suatu barisan bilangan Barisan Aritmatika adalah suatu barisan dengan selisih (beda) dua suku yang berurutan selalu tetap Bentuk Umum : U1, U2, U3, …., Un a, a + b, a + 2b,…., a + (n-1)b Pada barisan aritmatika,berlaku Un – Un-1 = b sehingga Un = Un-1 + b Hal.: 27 BARISAN DAN DERET 27

ARITHMETIC SEQUENCE AND SERIES 07/11/2017 ARITHMETIC SEQUENCE AND SERIES The orderly numbers like in speedometer have the same difference for every two orderly term, so it forms a sequence Arithmetic sequence is sequence with difference two orderly term constant The general form is : U1, U2, U3, …., Un a, a + b, a + 2b,…., a + (n-1)b In arithmetic sequence, we have Un – Un-1 = b, so Un = Un-1 + b Hal.: 28 BARISAN DAN DERET 28

BARISAN DAN DERET ARITMATIKA 07/11/2017 BARISAN DAN DERET ARITMATIKA Hal.: 29 BARISAN DAN DERET 29

ARITHMETIC SEQUENCE AND SERIES 07/11/2017 ARITHMETIC SEQUENCE AND SERIES If you start arithmetic sequence with the first term a and difference b, then you will get this following sequence The n-th term of arithmetic sequence is Un = a + ( n – 1 )b Where Un = n-th term a = the first term b = difference n = the term’s quantity a a + b a + 2b a + 3b …. a + (n-1)b Hal.: 30 BARISAN DAN DERET 30

BARISAN DAN DERET ARITMATIKA 07/11/2017 BARISAN DAN DERET ARITMATIKA Hl.: 31 Hal.: 31 BARISAN DAN DERET 31

ARITHMETIC SEQUENCE AND SERIES 07/11/2017 ARITHMETIC SEQUENCE AND SERIES If every term of arithmetic sequence is added, then we will get arithmetic series. Arithmetic series is the sum of terms of arithmetic sequence General form : U1 + U2 + U3 + … + Un atau a + (a +b) + (a+2b) +… + (a+(n-1)b) The formula of the sum of the first term in arithmetic series is Where S = the sum of n-th term n = the quantity of term a = the first term b = difference = n-th term Hal.: 32 BARISAN DAN DERET 32

BARISAN DAN DERET ARITMATIKA 07/11/2017 BARISAN DAN DERET ARITMATIKA Hal.: 33 BARISAN DAN DERET 33

ARITHMETIC SEQUENCE AND SERIES 07/11/2017 ARITHMETIC SEQUENCE AND SERIES Known: the sequence of 5, -2, -9, -16,…., find: The formula of n-th term The 25th term Answer: The difference of two orderly terms in sequence 5,-2, -9,-16 ,…is constant, b= -7, so that the sequence is an arithmetic sequence The formula of the n-th term in arithmetic sequence is Un = 5 + ( n – 1 ). -7 Un = 5 + - 7n + 7 Un = -7n + 12 b. The 25th term of arithmetic sequence is : U12 = - 7.12 + 12 = - 163 Hal.: 34 BARISAN DAN DERET 34

BARISAN DAN DERET GEOMETRI 07/11/2017 BARISAN DAN DERET GEOMETRI Barisan geometri adalah suatu barisan dengan pembanding (rasio) antara dua suku yang berurutan selalu tetap. Ada selembar kertas biru, akan dipotong-potong menjadi dua bagian. Hal.: 35 BARISAN DAN DERET 35

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES Geometric sequence is a sequence which has the constant ratio between two orderly term There is blue paper. It will cut into two pieces Hal.: 36 BARISAN DAN DERET 36

BARISAN DAN DERET GEOMETRI 07/11/2017 BARISAN DAN DERET GEOMETRI Hal.: 37 BARISAN DAN DERET 37

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES Look at the paper part that form a sequence 1 2 4 U1 U2 U3 Every two orderly terms of the sequence have the same ratio It seems that the ratio of every two orderly terms in the sequence is always constant. The sequence like this is called geometric sequence and the comparison of every two orderly term is called ratio (r) Hal.: 38 BARISAN DAN DERET

BARISAN DAN DERET GEOMETRI 07/11/2017 BARISAN DAN DERET GEOMETRI Hal.: 39 Hal.: 39 BARISAN DAN DERET 39

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES Geometric sequence is a sequence which have constant ratio for two orderly term General form: U1, U2, U3, …., Un atau a, ar, ar2, …., arn-1 In geometric sequence If you start the geometric sequence with the first term a and the ratio is r, then you get the following sequence Hal.: 40 BARISAN DAN DERET 40

BARISAN DAN DERET GEOMETRI 07/11/2017 BARISAN DAN DERET GEOMETRI Suku ke-n barisan Geometri adalah : Hal.: 41 BARISAN DAN DERET 41

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES Start With the first term a Multiply with ratio r Write the multiplication result The n-th term of geometric sequence is : Hal.: 42 BARISAN DAN DERET 42

BARISAN DAN DERET GEOMETRI 07/11/2017 BARISAN DAN DERET GEOMETRI Hubungan suku-suku barisan geometri Seperti dalam barisan Aritmatika hubungan antara suku yang satu dan suku yang lain dalam barisan geometri dapat dijelaskan sebagai berikut: Ambil U12 sebagai contoh : U12 = a.r11 U12 = a.r9.r2 = U10. r2 U12 = a.r8.r3 = U9. r3 U12 = a.r4.r7 = U5. r7 U12 = a.r3.r8 = U4.r8 Secara umum dapat dirumuskan bahwa : Un = Uk. rn-k Hal.: 43 BARISAN DAN DERET 43

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES The relation of terms in geometric sequence Like in arithmetic sequence, the relation between terms in geometric sequence can be explained as follows: Take U12 as example : U12 = a.r11 U12 = a.r9.r2 = U10. r2 U12 = a.r8.r3 = U9. r3 U12 = a.r4.r7 = U5. r7 U12 = a.r3.r8 = U4.r8 Generally, it can be formulated Un = Uk. rn-k Hal.: 44 BARISAN DAN DERET 44

BARISAN DAN DERET GEOMETRI 07/11/2017 BARISAN DAN DERET GEOMETRI Hal.: 45 BARISAN DAN DERET 45

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES Geometric series is the sum of terms in geometric sequence General form U1 + U2 + U3 + …. + Un a + ar + ar2 + ….+ arn-1 The formula of the n sum of the first term in geometric series is Hal.: 46 BARISAN DAN DERET 46

BARISAN DAN DERET GEOMETRI 07/11/2017 BARISAN DAN DERET GEOMETRI Hal.: 47 BARISAN DAN DERET 47

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES Known sequence 27, 9, 3, 1, …..find a.The formula of the n-th term b. The 8th term Answer: The ratio of two orderly terms in sequence 27,9,3, 1, …is constant, so that the sequence is a geometric sequence a. The formula of the n-th term in geometric sequence is Hal.: 48 BARISAN DAN DERET 48

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES b. The 8th term of geometric sequence is Hal.: 49 BARISAN DAN DERET

Deret Geometri tak hingga 07/11/2017 BARISAN DAN DERET GEOMETRI Deret Geometri tak hingga Deret geometi tak hingga adalah deret geometri yang banyak suku-sukunya tak hingga. Jika deret geometri tak hingga dengan -1 < r < 1 , maka jumlah deret geometri tak hingga tersebut mempunyai limit jumlah (konvergen). Untuk n = ∞ , rn mendekati 0 Sehingga S∞ = Dengan S∞ = Jumlah deret geometri tak hingga a = Suku pertama r = rasio Jika r < -1 atau r > 1 , maka deret geometri tak hingganya akan divergen, yaitu jumlah suku-sukunya tidak terbatas Hal.: 50 BARISAN DAN DERET 50

Infinite Geometric Series 07/11/2017 GEOMETRIC SEQUENCE AND SERIES Infinite Geometric Series Infinite geometric series is a geometric series which has infinite terms. If infinite geometric series is -1 < r < 1 , then the sum of geometric series has sum limit (convergent). For n = ∞ , rn is close to 0 So S∞ = With S∞ = the sum of infinite geometric series a = the first term r = ratio If r < -1 or r > 1 , then the infinite geometric series will be divergent, means the sum of terms is not limited Hal.: 51 BARISAN DAN DERET 51

BARISAN DAN DERET GEOMETRI 07/11/2017 BARISAN DAN DERET GEOMETRI Contoh : 1. Hitung jumlah deret geometri tak hingga : 18 + 6 + 2 + … . . Jawab : a = 18 ; Hal.: 52 BARISAN DAN DERET 52

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES Example : 1. Find the sum of infinite geometric series : 18 + 6 + 2 + … . . Answer : a = 18 ; Hal.: 53 BARISAN DAN DERET 53

BARISAN DAN DERET GEOMETRI 07/11/2017 BARISAN DAN DERET GEOMETRI 2. Sebuah bola elastis dijatuhkan dari ketinggian 2m. Setiap kali memantul dari lantai, bola mencapai ketinggian ¾ dari ketinggian sebelumnya. Berapakah panjang lintasan yang dilalui bola hingga berhenti ? Lihat gambar di samping! Bola dijatuhkan dari A, maka AB dilalui satu kali, selanjutnya CD, EF dan seterusnya dilalui dua kali. Lintasannya membentuk deret geometri dengan a = 3 dan r = ¾ Panjang lintasan = 2 S∞ - a = 14 Jadi panjang lintasan yang dilalui bola adalah14 m Hal.: 54 BARISAN DAN DERET 54

GEOMETRIC SEQUENCE AND SERIES 07/11/2017 GEOMETRIC SEQUENCE AND SERIES 2. An elastic ball is drop from the height of 2m. Every time it bounce from the floor, it has ¾ of the previous height. How long is the route that will be passed by the ball until it stop? Look at the picture! The ball is drop from A, so AB is passed only once. Then CD, EF, etc is passed twice. The route is in geometric series with a = 3 and r = ¾ the length of the route is= 2 S∞ - a = 14 So, the route length that pass by the ball is 14 m Hal.: 55 BARISAN DAN DERET 55

07/11/2017 TERIMA KASIH Hal.: 56 BARISAN DAN DERET 56

07/11/2017 THANK YOU Hal.: 57 BARISAN DAN DERET 57