Soal Matematika SD SMP SMA SMK UTS UAS UKK USBN: Induksi

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<b><a href="https://kuncijawabansoalmatematika.blogspot.com/" target="_blank">Soal Ulangan Harian Induksi Matematika SMA Kelas XII Kurikulum 2013 dan Pembahasannya</a></b> ini akan kami bagikan untuk menambah referensi para pengajar dalam mengumpulkan soal ulangan harian/Penilaian Harian, khususnya untuk jenjang SMA/MA Kelas 12. <b>Induksi Matematika</b> merupakan suatu teknik yang dikembangkan untuk membuktikan pernyataan. Induksi Matematika digunakan untuk mengecek hasil proses yang terjadi secara berulang sesuai dengan pola tertentu.</div>
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Indukasi Matematika digunakan untuk membuktikan universal statements " n Î A   S(n) dengan A Ì N  dan N adalah himpunan bilangan positif atau himpunan bilangan asli. S(n) adalah fungsi propositional.</div>
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Berikut ini contoh <a href="https://kuncijawabansoalmatematika.blogspot.com/2017/12/soal-ulangan-harian-induksi-matematika-sma-kelas-xii.html" target="_blank">Soal Ulangan Harian Induksi Matematika SMA Kelas XII Kurikulum 2013 dan Pembahasannya: </a></div>
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<span lang="IN" style="border: 0px; font-stretch: inherit; font-variant-numeric: inherit; line-height: 30px; margin: 0px; padding: 0px; vertical-align: baseline;"><b>Contoh 1.</b> Gunakan induksi matematik untuk membuktikan bahwa jumlah n buah bilangan ganjil positif pertama adalah<span style="color: #555555; font-family: Times New Roman, serif;"><span style="background-color: white; font-size: 12pt; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;"> </span></span><i style="background-color: white; border: 0px; color: rgb(85, 85, 85) !important; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">n</i><span style="background-color: white; border: 0px; color: rgb(85, 85, 85) !important; font-family: inherit; font-size: 13.76px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; position: relative; top: -0.4em; vertical-align: baseline;">2</span><span style="color: #555555; font-family: Times New Roman, serif;"><span style="background-color: white; font-size: 12pt; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;">.</span></span></span></div>
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Penyelesaian:</div>
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<i style="background-color: white; border: 0px; color: #333333; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><span lang="IN" style="border: 0px; color: rgb(85, 85, 85) !important; font-family: "Times New Roman", serif; font-size: 12pt; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 30px; margin: 0px; padding: 0px; vertical-align: baseline;">(i) <span style="border: 0px; font-family: "open sans", serif !important; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 30px !important; margin: 0px; padding: 0px; vertical-align: baseline;">   </span></span></i><span lang="IN" style="border: 0px; font-stretch: inherit; font-variant-numeric: inherit; line-height: 30px; margin: 0px; padding: 0px; vertical-align: baseline;">Basis induksi: Untuk n = 1, jumlah satu buah bilangan ganjil positif pertama adalah<span style="color: #555555; font-family: Times New Roman, serif;"><span style="background-color: white; font-size: 12pt; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;"> 1</span></span><span style="background-color: white; border: 0px; color: rgb(85, 85, 85) !important; font-family: inherit; font-size: 13.76px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; position: relative; top: -0.4em; vertical-align: baseline;">2</span><span style="color: #555555; font-family: Times New Roman, serif;"><span style="background-color: white; font-size: 12pt; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;"> </span></span></span>= 1.          Ini benar karena jumlah satu buah bilangan ganjil positif pertama adalah 1.</div>
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<i style="background-color: white; border: 0px; color: #333333; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><span lang="IN" style="border: 0px; color: rgb(85, 85, 85) !important; font-family: "Times New Roman", serif; font-size: 12pt; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 30px; margin: 0px; padding: 0px; vertical-align: baseline;">(ii) <span style="border: 0px; font-family: "open sans", serif !important; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 30px !important; margin: 0px; padding: 0px; vertical-align: baseline;">  </span></span></i>Langkah induksi: Andaikan p(n) benar, yaitu pernyataan</div>
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        1 + 3 + 5 + … + (2n – 1) =<span style="background-color: white; color: #555555; font-family: "Times New Roman", serif; font-size: 12pt; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;"> </span><i style="background-color: white; border: 0px; color: #555555; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">n</i><span style="background-color: white; border: 0px; color: #555555; font-family: inherit; font-size: 13.76px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; position: relative; top: -0.4em; vertical-align: baseline;">2</span></div>
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adalah benar (hipotesis induksi) [catatlah bahwa bilangan ganjil positif ke-n adalah (2n– 1)]. Kita harus memperlihatkan bahwa p(n +1) juga benar, yaitu</div>
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<span lang="IN" style="border: 0px; font-stretch: inherit; font-variant-numeric: inherit; line-height: 30px; margin: 0px; padding: 0px; vertical-align: baseline;">1 + 3 + 5 + … + (2n – 1) + (2n + 1) = (n + 1<span style="color: #555555; font-family: Times New Roman, serif;"><span style="background-color: white; font-size: 12pt; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;">)</span></span><span style="background-color: white; border: 0px; color: rgb(85, 85, 85) !important; font-family: inherit; font-size: 13.76px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; position: relative; top: -0.4em; vertical-align: baseline;">2</span></span></div>
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juga benar. Hal ini dapat kita tunjukkan sebagai berikut:</div>
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1 + 3 + 5 + … + (2n – 1) + (2n + 1)</div>
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= [1 + 3 + 5 + … + (2n – 1)] + (2n +1)</div>
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<span lang="IN" style="border: 0px; color: rgb(85, 85, 85) !important; font-family: "Times New Roman", serif; font-size: 12pt; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 30px; margin: 0px; padding: 0px; vertical-align: baseline;">= <i style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">n</i><span style="border: 0px; font-family: inherit; font-size: 13.76px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; position: relative; top: -0.4em; vertical-align: baseline;">2</span> + (2<i style="border: 0px; font-family: inherit; font-size: inherit; font-stretch: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">n</i> + 1)</span></div>
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<span lang="IN" style="border: 0px; font-stretch: inherit; font-variant-numeric: inherit; line-height: 30px; margin: 0px; padding: 0px; vertical-align: baseline;">Karena langkah basis dan langkah induksi keduanya telah diperlihatkan benar, maka jumlah n buah bilangan ganjil positif pertama adalah<span style="color: #555555; font-family: Times New Roman, serif;"><span style="background-color: white; font-size: 12pt; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;"> </span></span><i style="background-color: white; border: 0px; color: rgb(85, 85, 85) !important; font-family: inherit; font-size: inherit; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">n</i><span style="background-color: white; border: 0px; color: rgb(85, 85, 85) !important; font-family: inherit; font-size: 13.76px; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: inherit; margin: 0px; padding: 0px; position: relative; top: -0.4em; vertical-align: baseline;">2</span><span style="color: #555555; font-family: Times New Roman, serif;"><span style="background-color: white; font-size: 12pt; font-style: inherit; font-variant-caps: inherit; font-variant-ligatures: inherit; font-weight: inherit;">.</span></span><span style="background-color: white; border: 0px; color: rgb(85, 85, 85) !important; font-family: "open sans", serif !important; font-size: 12pt; font-stretch: inherit; font-style: inherit; font-variant: inherit; font-weight: inherit; line-height: 30px !important; margin: 0px; padding: 0px; vertical-align: baseline;">  </span></span></div>
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Untuk lebih lengkapnya, silakan download <b>Soal Ulangan Harian Induksi Matematika SMA Kelas XII Kurikulum 2013 dan Pembahasannya</b> [<b><a href="https://drive.google.com/open?id=136-hFZdXn2uSrHybUBy3yhQJj5s1KxKa" target="_blank">DI SINI</a></b>]</div>

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