LKPD INDUKSI Matematika [PDF]

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LEMBAR KERJA PESERTA DIDIK Materi Induksi Matematika Mata Pelajaran Nama Siswa Kelas/Semester



: Matematika : ...................................................................................................................... : XI Umum/Ganjil



Kompetensi Dasar (KD) 3.1 Menjelaskan metode pembuktian pernyataan matematis berupa barisan, ketidaksamaan, keterbagian dengan induksi matematika. Indikator Pencapaian Kompetensi (IPK) 3.1.3 Membuktikan formula suatu barisan bilangan dengan prinsip induksi matematika Materi Misalkan P(n) merupakan suatu pernyataan bilangan asli. Pernyataan P(n) benar jika memenuhi langkah berikut ini: a. Langkah Awal (Basic Step): Buktikan P(n) benar untuk n=1 b. Langkah Induksi (Induction Step): Asumsikan P(n) benar untuk n = k, lalu buktikan P(n) benar untuk n = k + 1 Contoh: 1 Buktikan bahwa 1+2+3+...+ n = 2 𝑛(𝑛 + 1) ! Jawaban: Langkah 1: Akan dibuktikan P(n) benar untuk n = 1 1



1 + 2 + 3 + ⋯ + 𝑛 = 2 𝑛(𝑛 + 1) 1



𝑛 = 2 𝑛(𝑛 + 1)



ganti n dengan 1



1



1 = 2 . 1(1 + 1) 1



1 = 2 (2) 1 = 1 (ruas kiri = ruas kanan) Oleh karena ruas kiri = ruas kanan, maka P(n) benar untuk n = 1 Langkah 2: Diasumsikan P(n) benar, untuk n = k 1



1 + 2 + 3 + ⋯ + 𝑛 = 2 𝑛(𝑛 + 1)



ganti n dengan k



1



1 + 2 + 3 + ⋯ + 𝑘 = 2 𝑘(𝑘 + 1) Akan dibuktikan P(n) benar untuk n = k+1 1



1 + 2 + 3 + ⋯ + 𝑘 = 2 𝑘(𝑘 + 1)



1



1 + 2 + 3 + ⋯ + 𝑘 + (𝑘 + 1) = 2 (𝑘 + 1)(𝑘 + 1 + 1) Hasil pada Langkah 2 1 𝑘(𝑘 + 1) 2 1 2 1 𝑘 +2𝑘 2 1 2



1



+(𝑘 + 1) = 2 (𝑘 + 1)(𝑘 + 1 + 1)



Ruas kiri ditambah pengulangan suku terakhir (k), dimana k nya ditambah 1, ruas kanan setiap k diubah ke k+1



1



+ 𝑘 + 1 = 2 (𝑘 + 1)(𝑘 + 2) 3



1



𝑘 2 + 2 𝑘 + 1 = 2 (𝑘 2 + 2𝑘 + 𝑘 + 1)



1 2 1



3



1



3



1



𝑘 2 + 2 𝑘 + 1 = 2 (𝑘 2 + 3𝑘 + 1) 3



𝑘 2 + 2 𝑘 + 1 = 2 𝑘 2 + 2 𝑘 + 1 (ruas kiri = ruas kanan) Oleh karena ruas kiri = ruas kanan, maka P(n) benar untuk n = k+1 2



Kesimpulan: Oleh karena Langkah 1 dan Langkah 2 bernilai benar, maka TERBUKTI bahwa 1 1+2+3+...+ n = 2 𝑛(𝑛 + 1)



Rumusan Soal Petunjuk: Ikutilah langkah-langkah pada contoh di atas untuk menyelesaikan soal-soal di bawah ini! 1. Buktikan bahwa 3+5+7+9+...+ (2n+1) = (𝑛2 + 2𝑛) ! Jawaban Langkah 1: Akan dibuktikan P(n) benar untuk n = 1 ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... Langkah 2: Diasumsikan P(n) benar, untuk n = k ..................................................................................................................................................... ..................................................................................................................................................... Akan dibuktikan P(n) benar untuk n = k+1 ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... Kesimpulan:................................................................................................................................. ..................................................................................................................................................... 1 2. Buktikan bahwa 8+11+14+17+...+ (3n+5) = 2 𝑛(3𝑛 + 13) ! Jawaban Langkah 1: Akan dibuktikan P(n) benar untuk n = 1 ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... Langkah 2: Diasumsikan P(n) benar, untuk n = k ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... Akan dibuktikan P(n) benar untuk n = k+1 ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... Kesimpulan:................................................................................................................................. .....................................................................................................................................................