{"id":252551,"date":"2019-05-10T14:51:20","date_gmt":"2019-05-10T07:51:20","guid":{"rendered":"https:\/\/www.quipper.com\/id\/blog\/?p=252551"},"modified":"2021-11-24T17:56:07","modified_gmt":"2021-11-24T10:56:07","slug":"induksi-matematika-kelas-12","status":"publish","type":"post","link":"https:\/\/quipperhome.wpcomstaging.com\/mapel\/induksi-matematika-kelas-12\/","title":{"rendered":"Buat yang Anak Kelas 12, Yuk Belajar Induksi Matematika beserta Contoh Soalnya!"},"content":{"rendered":"<p><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-252731\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya2.png\" alt=\"\" width=\"800\" height=\"533\"><\/p>\n<blockquote><p><span style=\"font-weight: 400;\">Induksi matematika merupakan suatu metode untuk membuktikan bahwa suatu pernyataan tertentu berlaku untuk setiap bilangan asli. Ada beberapa jenis: induksi matematika sederhana, induksi matematika diperluas, dan induksi matematika kuat.<\/span><\/p><\/blockquote>\n<p><span style=\"font-weight: 400;\">Hai Quipperian! Jika ditanya mata pelajaran apa yang kalian sukai, kira-kira jawabannya apa? Apakah ada di antara kalian yang menyukai mata pelajaran Matematika? Membahas Matematika sama artinya membahas bahasa melalui hitungan. Itulah mengapa beberapa dari Quipperian mungkin tidak begitu suka dengan Matematika. <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Sejatinya, Matematika itu mudah jika Quipperian mau terus berlatih dan membaca. Selain itu, Matematika juga ilmu yang erat dengan kehidupan sehari-hari. Contohnya saat Quipperian mengambil uang melalui Anjungan Tunai Mandiri (ATM). Bagaimana bisa sistem dalam ATM multi nominal, mengetahui batas minimal pengambilan <\/span><em><span style=\"font-weight: 400;\">user<\/span><\/em><span style=\"font-weight: 400;\">? Ternyata, di dalam sistem ATM multi nominal berlaku penerapan induksi matematika, <\/span><em><span style=\"font-weight: 400;\">lho<\/span><\/em><span style=\"font-weight: 400;\">. Terus, apakah induksi matematika itu?<\/span><\/p>\n<h2><strong>Pengertian Induksi Matematika<\/strong><\/h2>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-252721\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya3.png\" alt=\"\" width=\"800\" height=\"533\"><\/p>\n<p><span style=\"font-weight: 400;\">Induksi matematika merupakan suatu metode untuk membuktikan bahwa suatu pernyataan tertentu berlaku untuk setiap bilangan asli. Induksi matematika ini merupakan metode baku dalam pembuktian di bidang Matematika. Dengan adanya Induksi matematika ini, Quipperian bisa meminimalisir langkah-langkah untuk membuktikan bahwa semua bilangan bulat termasuk dalam himpunan kebenaran.<\/span><\/p>\n<h2><strong>Induksi Matematika Sederhana<\/strong><\/h2>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-145732\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2017\/10\/Latihan-Soal-Cerita-Matematika-untuk-Tes-STAN-Yuk.jpg\" alt=\"Latihan Soal Cerita Matematika untuk Tes STAN, Yuk!\" width=\"800\" height=\"533\"><\/p>\n<p><span style=\"font-weight: 400;\">Induksi Matematika memiliki langkah dasar yang harus ditempuh untuk membuktikan bahwa kebenaran suatu pernyataan berlaku untuk setiap bilangan asli. Adapun langkah-langkahnya adalah sebagai berikut.<\/span><\/p>\n<ol>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Langkah dasar: Pada langkah ini, Quipperian harus membuktikan bahwa suatu pernyataan berlaku untuk P(1) atau P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\">).<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Langkah induksi: Jika suatu pernyataan berlaku untuk P(1) atau P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\">), maka pernyataan itu juga harus berlaku untuk p(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">) atau P(<\/span><em><span style=\"font-weight: 400;\">k <\/span><\/em><span style=\"font-weight: 400;\">+ 1).<\/span><\/li>\n<\/ol>\n<p><strong>Contoh Soal<\/strong><\/p>\n<p><span style=\"font-weight: 400;\">Buktikan bahwa penjumlahan <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\"> bilangan asli berurutan berlaku!<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-252691\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/induksi-matematika-no-1-300x61.png\" alt=\"\" width=\"300\" height=\"61\" srcset=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-1-300x61.png 300w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-1-768x156.png 768w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-1-1024x208.png 1024w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-1-400x81.png 400w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-1.png 1999w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><strong>Pembahasan:<\/strong><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Pertama-tama, kamu harus menentukan langkah dasarnya.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Langkah dasar:<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-252671\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/induksi-matematika-no-2-300x92.png\" alt=\"\" width=\"300\" height=\"92\" srcset=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-2-300x92.png 300w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-2-768x236.png 768w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-2-1024x315.png 1024w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-2-293x90.png 293w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-2.png 1802w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">Oleh karena P(1) = 1, maka jelas benar (berlaku), artinya P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><sub><em><span style=\"font-weight: 400;\">o<\/span><\/em><\/sub><span style=\"font-weight: 400;\">) = benar<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Langkah induksi: Jika P(1) benar, maka pernyataan tersebut harus benar untuk P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">+1) dengan <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> \u2265 <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><sub><em><span style=\"font-weight: 400;\">o<\/span><\/em><\/sub><span style=\"font-weight: 400;\">,<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\"> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-252681\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/induksi-matematika-no-3-300x62.png\" alt=\"\" width=\"300\" height=\"62\">benar<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Sehingga:<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-252651\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/induksi-matematika-no-4-300x176.png\" alt=\"\" width=\"300\" height=\"176\"><\/p>\n<p><span style=\"font-weight: 400;\">Oleh karena P(<\/span><em><span style=\"font-weight: 400;\">k <\/span><\/em><span style=\"font-weight: 400;\">+ 1) mengikuti bentuk pernyataan P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\">), maka P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">) bernilai benar.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Bagaimana Quipperian, apakah kalian sudah memahami penyelesaian soal di atas? Pada prinsipnya, induksi matematika hanyalah metode untuk membuktikan bahwa pernyataan <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-252661\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/induksi-matematika-no-5-300x62.png\" alt=\"\" width=\"300\" height=\"62\" srcset=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-5-300x62.png 300w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-5-768x159.png 768w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-5-1024x212.png 1024w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-5-400x83.png 400w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-5.png 1999w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/span><span style=\"font-weight: 400;\">bernilai benar untuk semua bilangan <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\"> \u2265 1.<\/span><\/p>\n<h2><strong>Induksi Matematika Diperluas<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-252741\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya1.png\" alt=\"\" width=\"800\" height=\"533\" srcset=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya1.png 800w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya1-300x200.png 300w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya1-768x512.png 768w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya1-135x90.png 135w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">Setiap pernyataan yang memuat <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\"> bilangan asli, ternyata tidak harus dimulai dari angka 1, <\/span><em><span style=\"font-weight: 400;\">lho<\/span><\/em><span style=\"font-weight: 400;\">. Itulah sebabnya, Induksi Matematika bisa diperluas dengan langkah-langkah berikut.<\/span><\/p>\n<ol>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Langkah dasar: Pembuktian bahwa suatu pernyataan berlaku untuk P(<\/span><em><span style=\"font-weight: 400;\">m<\/span><\/em><span style=\"font-weight: 400;\">).<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Langkah induksi: Pembuktian bahwa jika pernyataan berlaku untuk P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">), dengan <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">\u2265<\/span><em><span style=\"font-weight: 400;\">m<\/span><\/em><span style=\"font-weight: 400;\">, maka pernyataan tersebut juga berlaku untuk P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> + 1).<\/span><\/li>\n<\/ol>\n<p><strong>Contoh Soal<\/strong><\/p>\n<p><span style=\"font-weight: 400;\">Buktikan bahwa <em>n<sup>2<\/sup>\u2265 2n + 7<\/em><\/span><span style=\"font-weight: 400;\">&nbsp;untuk semua bilangan asli <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\">\u22655!<\/span><\/p>\n<p><strong>Pembahasan:<\/strong><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Quipperian harus memisalkan bahwa P(<em>n<\/em>) =&nbsp;<em>n<sup>2<\/sup>\u2265 2n +&nbsp;<\/em>7<\/span><span style=\"font-weight: 400;\">&nbsp;untuk semua bilangan asli <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\">\u22654.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Bilangan asli yang bisa Quipperian masukkan di awal, bukanlah 1 melainkan 4 karena terdapat keterangan bahwa semua bilangan asli <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\">\u22654.<\/span><\/p>\n<p>P(4) = 4<sup>2<\/sup>\u2265 2(4) + 7<\/p>\n<p><em>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=&nbsp;<\/em>16 <em>\u2265&nbsp;<\/em>15<\/p>\n<p><span style=\"font-weight: 400;\">Artinya, P(4) bernilai benar<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Selanjutnya, Quipperian harus memisalkan bahwa P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">) benar untuk <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">\u22654 (hipotesis induksi).<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Berdasarkan hipotesis di atas diperoleh:<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-252621\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/induksi-matematika-no-9-300x126.png\" alt=\"\" width=\"300\" height=\"126\" srcset=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-9-300x126.png 300w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-9-768x323.png 768w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-9-1024x430.png 1024w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-9-214x90.png 214w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-9.png 1542w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">Setelah Quipperian mendapatkan persamaan di atas, cobalah buktikan bahwa P(<\/span><em><span style=\"font-weight: 400;\">k <\/span><\/em><span style=\"font-weight: 400;\">+ 1) juga bernilai benar.<\/span><\/p>\n<p>(<em>k<\/em>+1)<sup>2<\/sup>\u2265 2(<em>k<\/em>+1) + 7<\/p>\n<p><span style=\"font-weight: 400;\">Hipotesis induksi menyatakan bahwa (<em>k<\/em>+1)<sup>2<\/sup>\u2265 2(<em>k<\/em>+1) + 14&nbsp;<\/span><span style=\"font-weight: 400;\">bernilai benar, maka (<em>k<\/em>+1)<sup>2<\/sup>\u2265 2(<em>k<\/em>+1) + 7&nbsp;<\/span><span style=\"font-weight: 400;\">otomatis juga akan bernilai benar karena 14 &gt; 7.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Jadi, pernyataan <em>n<sup>2<\/sup>\u2265 2n + 7<\/em>&nbsp;<\/span><span style=\"font-weight: 400;\"> benar untuk setiap <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\">\u22654.<\/span><\/p>\n<h2><strong>Induksi Matematika Kuat<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-252751\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya.png\" alt=\"\" width=\"800\" height=\"533\" srcset=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya.png 800w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya-300x200.png 300w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya-768x512.png 768w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/Buat-yang-Anak-Kelas-12-Yuk-Belajar-Induksi-Matematika-beserta-Contoh-Soalnya-135x90.png 135w\" sizes=\"(max-width: 800px) 100vw, 800px\" \/><\/p>\n<p><span style=\"font-weight: 400;\">Prinsip dasar pada induksi matematika kuat ini berbeda dengan sebelumnya. Jika sebelumnya Quipperian cukup membuktikan bahwa P(1) benar, maka pada induksi matematika kuat ini, pernyataan harus bernilai benar untuk P(1), P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><sub><em><span style=\"font-weight: 400;\">o<\/span><\/em><\/sub><span style=\"font-weight: 400;\"> + 1), P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><sub><em><span style=\"font-weight: 400;\">o<\/span><\/em><\/sub><span style=\"font-weight: 400;\"> + 2), \u2026, P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">). Selain itu, Quipperian juga harus membuktikan pernyataan benar untuk P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> + 1). Berikut ini adalah langkah-langkah yang harus Quipperian tempuh untuk induksi matematika kuat.<\/span><\/p>\n<ol>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Langkah dasar: Buktikan bahwa P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><sub><em><span style=\"font-weight: 400;\">o<\/span><\/em><\/sub><span style=\"font-weight: 400;\">) benar.<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Langkah induksi: Jika P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><sub><em><span style=\"font-weight: 400;\">o<\/span><\/em><\/sub><span style=\"font-weight: 400;\">), P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><sub><em><span style=\"font-weight: 400;\">o<\/span><\/em><\/sub><span style=\"font-weight: 400;\"> + 1), P(<\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><sub><em><span style=\"font-weight: 400;\">o<\/span><\/em><\/sub><span style=\"font-weight: 400;\"> + 2), \u2026, P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">) benar untuk <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> \u2265 <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><sub><em><span style=\"font-weight: 400;\">o<\/span><\/em><\/sub><span style=\"font-weight: 400;\">, maka gunakan hal itu untuk membuktikan bahwa P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> + 1) juga benar.<\/span><\/li>\n<\/ol>\n<p><strong>Contoh Soal<\/strong><\/p>\n<p><span style=\"font-weight: 400;\">Buktikan bahwa setiap bilangan bulat positif <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\"> \u2265 2 dapat dinyatakan sebagai perkalian dari satu atau lebih bilangan prima.<\/span><\/p>\n<p><strong>Pembahasan:<\/strong><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Quipperian harus menentukan langkah dasarnya terlebih dahulu, yaitu&nbsp;<em>n = 2&nbsp;<\/em><\/span>dengan 2 merupakan bilangan prima, sehingga pernyataan ini benar.<\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Quipperian bisa melanjutkan dengan menentukan langkah induksinya.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Coba Quipperian misalkan bahwa 2, 3, 4, 5, \u2026, <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> dapat dinyatakan sebagai hasil perkalian antara satu atau lebih bilangan prima. Dengan demikian, <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> +1 juga merupakan hasil perkalian antara satu atau lebih bilangan prima. Artinya <\/span><em><span style=\"font-weight: 400;\">&nbsp;k <\/span><\/em><span style=\"font-weight: 400;\">+ 1 bisa berupa bilangan prima atau nonprima (komposit).<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Jika <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> + 1 merupakan bilangan prima, maka <\/span><em><span style=\"font-weight: 400;\">k <\/span><\/em><span style=\"font-weight: 400;\">+ 1 dapat dinyatakan sebagai hasil kali <\/span><em><span style=\"font-weight: 400;\">k <\/span><\/em><span style=\"font-weight: 400;\">&nbsp;+ 1 itu sendiri.<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Jika <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> + 1 bukan bilangan prima, maka pembagi <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> + 1 tidak hanya 1 atau <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> + 1 itu sendiri, melainkan ada bilangan lain. Misalnya, bilangan lain itu dinotasikan <\/span><em><span style=\"font-weight: 400;\">p<\/span><\/em><span style=\"font-weight: 400;\"> dan hasil baginya <\/span><em><span style=\"font-weight: 400;\">q<\/span><\/em><span style=\"font-weight: 400;\">. Secara matematis, dapat ditulis sebagai berikut.<\/span><\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-252571 aligncenter\" src=\"https:\/\/quipper-video-wordpress.s3.amazonaws.com\/images\/2019\/05\/induksi-matematika-no-13-300x98.png\" alt=\"\" width=\"300\" height=\"98\" srcset=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-13-300x98.png 300w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-13-768x250.png 768w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-13-1024x333.png 1024w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-13-277x90.png 277w, https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/05\/induksi-matematika-no-13.png 1753w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Oleh karena 2 \u2264 <\/span><em><span style=\"font-weight: 400;\">p<\/span><\/em><span style=\"font-weight: 400;\">, <\/span><em><span style=\"font-weight: 400;\">q<\/span><\/em><span style=\"font-weight: 400;\"> \u2264 <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">, maka nilai <\/span><em><span style=\"font-weight: 400;\">p<\/span><\/em><span style=\"font-weight: 400;\"> dan <\/span><em><span style=\"font-weight: 400;\">q<\/span><\/em><span style=\"font-weight: 400;\"> yang mungkin, yaitu 2, 3, 4, \u2026, <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\">. Jika Quipperian perhatikan, nilai <\/span><em><span style=\"font-weight: 400;\">p<\/span><\/em><span style=\"font-weight: 400;\"> dan <\/span><em><span style=\"font-weight: 400;\">q<\/span><\/em><span style=\"font-weight: 400;\"> yang mungkin merupakan hasil kali satu atau lebih bilangan prima, sehingga <\/span><em><span style=\"font-weight: 400;\">pq<\/span><\/em><span style=\"font-weight: 400;\"> jelas menunjukkan hasil kali satu atau lebih bilangan prima. Oleh karena <em>k + 1 = pq,<\/em><\/span><span style=\"font-weight: 400;\">&nbsp;maka <\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> + 1 juga merupakan hasil kali satu atau lebih bilangan prima. Artinya, P(<\/span><em><span style=\"font-weight: 400;\">k<\/span><\/em><span style=\"font-weight: 400;\"> + 1) bernilai benar.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Jadi, terbukti ya Quipperian jika pernyataan <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\"> \u2265 2 benar untuk setiap bilangan asli <\/span><em><span style=\"font-weight: 400;\">n<\/span><\/em><span style=\"font-weight: 400;\"> \u2265 2.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Ternyata, belajar induksi matematika itu mudah kan, Quipperian? Jangan pernah menyerah untuk mempelajari Matematika karena itulah ilmu dasar yang dekat dengan kehidupan. Ingin belajar Matematika dengan suasana menyenangkan? Ayo, segera gabung dengan <\/span><a href=\"https:\/\/learn.quipper.com\/signup\"><strong>Quipper Video<\/strong><\/a><span style=\"font-weight: 400;\">. Temukan ratusan modul di dalamnya dan kerjakan soal-soalnya agar kalian bisa lebih paham. Salam Quipper!<\/span><\/p>\n<p><strong>Sumber:<\/strong><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><a href=\"https:\/\/www.m4th-lab.net\/2017\/07\/induksi-matematika-induksi-matematika.html\"><span style=\"font-weight: 400;\">https:\/\/www.m4th-lab.net\/2017\/07\/induksi-matematika-induksi-matematika.html<\/span><\/a><\/li>\n<li style=\"font-weight: 400;\"><a href=\"http:\/\/www.oasisilmu.com\/index.php\/2016\/11\/24\/mengaplikasikan-induksi-matematika-pada-atm\/\"><span style=\"font-weight: 400;\">http:\/\/www.oasisilmu.com\/index.php\/2016\/11\/24\/mengaplikasikan-induksi-matematika-pada-atm\/<\/span><\/a><\/li>\n<li style=\"font-weight: 400;\"><a href=\"http:\/\/adrahma2.blogspot.com\/2013\/04\/penerapan-induksi-matematik.html\"><span style=\"font-weight: 400;\">http:\/\/adrahma2.blogspot.com\/2013\/04\/penerapan-induksi-matematik.html<\/span><\/a><\/li>\n<li style=\"font-weight: 400;\"><a href=\"https:\/\/learn.quipper.com\/v1\/topic\/564e8a9da71a13001100843f\/documents\/564e9240a71a1300110089b3?token=UPC%2Fpl6b3R4lsHR52pp%2FL4rbs57h%2Fs1k486JmMJcaV2fmlQVnfcpr89eOowt7xDSzclaxp6K%2FV0QjIaQyIUNyQ%3D%3D\"><span style=\"font-weight: 400;\">https:\/\/learn.quipper.com\/v1\/topic\/564e8a9da71a13001100843f\/documents\/564e9240a71a1300110089b3?token=UPC%2Fpl6b3R4lsHR52pp%2FL4rbs57h%2Fs1k486JmMJcaV2fmlQVnfcpr89eOowt7xDSzclaxp6K%2FV0QjIaQyIUNyQ%3D%3D<\/span><\/a><\/li>\n<\/ul>\n<p><strong>Penulis: Eka Viandari<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Induksi matematika merupakan suatu metode untuk membuktikan bahwa suatu pernyataan tertentu berlaku untuk setiap bilangan asli. Ada beberapa jenis: induksi matematika sederhana, induksi matematika&hellip;<\/p>\n","protected":false},"author":99400369,"featured_media":252731,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_monsterinsights_skip_tracking":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_wpcom_ai_launchpad_first_post":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[679384888,679384865],"tags":[],"ppma_author":[679386827],"class_list":["post-252551","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mapel","category-matematika"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Buat yang Anak Kelas 12, Yuk Belajar Induksi Matematika beserta Contoh Soalnya!<\/title>\n<meta name=\"description\" content=\"Kamu merasa belum menguasai banget Matematika, Quipperian? 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