Induksi Matematika
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Induksi Matematika

Course Number: MIPA 9, Spring 2012

College/University: Universitas Gadjah Mada

Word Count: 630

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BAB IV INDUKSI MATEMATIKA Salah satu alat bukti penting yang banyak dipakai di bidang matematika, khususnya yang terkait dengan himpunan bilangan asli adalah induksi matematika yang sesungguhnya merupakan salah satu aksioma yang dipenuhi oleh system bilangan asli. Bentuk umum induksi matematika sebagai berikut: Misalkan adalah himpunan semua bilangan asli. Jika Teorema 4.1 1. Pangkal: subhimpunan yang...

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IV INDUKSI BAB MATEMATIKA Salah satu alat bukti penting yang banyak dipakai di bidang matematika, khususnya yang terkait dengan himpunan bilangan asli adalah induksi matematika yang sesungguhnya merupakan salah satu aksioma yang dipenuhi oleh system bilangan asli. Bentuk umum induksi matematika sebagai berikut: Misalkan adalah himpunan semua bilangan asli. Jika Teorema 4.1 1. Pangkal: subhimpunan yang memenuhi , dan 2. Induksi hipotesia: berakibat maka , . Selanjutnya jika sifat ” atau , dengan mempunyai arti mempunyai memenuhi ketentuan ”, maka berlaku . Hal ini berakibat Teorema 4.1 ekuivalen dengan Teorema 4.2 1. Pangkal: Jika subhimpunan yang memenuhi , dan 2. Induksi hipotesia: berakibat , maka Contoh 4.0.4 . Dengan menggunakan induksi matematika buktikan, bahwa: 1. Jumlah suku pertama deret geometri dengan rasio dan adalah 2. Jumlah suku pertama deret aritmatika dengan beda dan adalah . Penyelesaian: 1. Jumlah suku pertama deret geomeri dengan ! dibentuk # % , ! sehingga adalah: " $! 1. Pangkal: Untuk dan suku ke-1 ! . 2. Induksi hipotesa: Misalkan . Akibatnya " ! sehingga ! & " & & ' & () Jadi , sehingga berlaku bilangan asli berlaku ! item * + Jumlah . Dengan kata lain untuk setiap suku pertama deret hitung dengan beda dan suku pertama adalah ! " , Didefinisikan . mempunyai sifat ”, yaitu ! 1. Pangkal: Untuk ! ! - . , sehingga 1 memenuhi sifat . 2. Induksi hipotesa: Misalkan memenuhi sifat . Akibatnya " ! , sehingga ! & " - . - & Jadi . & memenuhi sifat , sehingga untuk setiap bilangan asli ! berlaku . Diketahui Contoh 4.2.7 dan /. adalah dua bilangan bulat positif dan Tentukan, berlaku akan sifat , yaitu dapat ditemukan bilangan bulat 0 1 2 dan yang memenuhi 2 4 3 5 , untuk 6 3 27 7 8 7 07 9 2, dan 9 9 Penyelesaian: 9 9 8 : / 2. Diketahui bilangan bulat 1. Pangkal: , dengan 0 2 dan 7 9 sehingga 1 memenuhi sifat . Induksi hipotesa: Misalkan 0 1 27 9 27 2 4 3 5 9 sehingga mempunyai sifat untuk 6 9 9 27 7 8 7 0 sedemikian hingga 9 8 9& : , memenuhi sifat . 2. Ada indeks 6 , 2 4 6 4 0 yang memenuhi 3 5 pertama yang memenuhi adalah ;. Akibatnya 2 4 9 9 Contoh 4.2.8 , berarti dapat ditemukan 9 9 9 8 Buktikan bahwa bulat positif . Bukti: 1. Pangkal: Untuk 9 <& 8 < 5 dan : <& = . Anggap saja indeks < " < 7 untuk setiap bilangan . sehingga 1 memenuhi sifat. 2. Induksi hipotesa: Misalkan = = memenuhi " . Maka " . Latihan 4.3 1. Dengan menggunakan induksi matematika buktikan bahwa untuk 7 78 persamaan dan pertidaksamaan ini berlaku = 1.1 ? 1.2 & " ? 8 ? & 1.3 " 1.4 = " AB " 1.5 > >A & ? & & & & & 4 1.6 4 1.7 C 1.8 1.9 @ > D 1.12 Jika C , untuk setiap C 1 . E;, untuk suatu bilangan asli ;. 1.10 E D 1.11 1 = C F;, untuk suatu bilangan asli ;. 1 C , untuk setiap C 1 . bulat positif, untuk 6 ) 8 G4 8 3 7 7 87 maka 2. Buktikan dengan induksi matematika, bahwa buah garis lurus pada sebuah bidang datar pasti membagi bidang tersebut menjadi G& & dareah, dengan asumsi tidak ada dua garis lurus yang sejajar dan tidak ada tiga garis yang beririsan di suatu titik. L 3. Tunjukkan bahwa jika H J K M I 2, maka HJ C I HJ C I " HJ C I HJ J I L N F HJ K M I O PHK L HJ K M I & CM

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Universitas Gadjah Mada - MIPA - 9
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