Contoh soal Induksi

August 21, 2018 cloudrain1 Uncategorized

Buktikan dengan menggunakan induksi matematika

$\displaystyle 1+3+5+...+(2n-1)+(2n-3)+...+3+1=2{{n}^{2}}-2n+1$

Bukti:

Untuk $n=2$ maka

$\begin{array}{l}1+3+1=2\cdot {{2}^{2}}-2\cdot 2+1\\5=5\end{array}$

Benar.

Andaikan pernyataan benar untuk $n=k,$ yaitu

$1+3+5+...+\left( {2k-1} \right)+\left( {2k-3} \right)+...+3+1=2{{k}^{2}}-2k+1$ benar.

Hendak dibuktikan pernyataan benar untuk $n=k+1,$ yaitu

$\begin{array}{l}1+3+5+...+\left( {2k-1} \right)+(2k+1)+(2k-1)+(2k-3)+...+3+1=2{{(k+1)}^{2}}-2(k+1)+1\\=2\left( {{{k}^{2}}+2k+1} \right)-2k-2+1\\=2{{k}^{2}}+4k+2-2k-1\\=2{{k}^{2}}+2k+1\end{array}$

Bukti:

$\begin{array}{l}1+3+5+...+(2k-1)+(2k+1)+(2k-1)+(2k-3)+...+3+1\\=2{{k}^{2}}-2k+1+\left( {2k+1} \right)+\left( {2k-1} \right)\\=2{{k}^{2}}+2k+1\end{array}$

Terbukti.

Pembuktian rumus penjumlahan bilangan kuadrat berurutan

August 3, 2018August 3, 2018 cloudrain1 Uncategorized

$\displaystyle \begin{array}{l}\text{Jelas bahwa }{{(k+1)}^{3}}-{{k}^{3}}=3{{k}^{2}}+3k+1\\\text{maka }\sum\limits_{{k=1}}^{n}{{({{{(k+1)}}^{3}}-{{k}^{3}})}}=3\sum\limits_{{k=1}}^{n}{{{{k}^{2}}}}+3\sum\limits_{{k=1}}^{n}{k}+\sum\limits_{{k=1}}^{n}{1}\\\text{Oleh karena }\sum\limits_{{k=1}}^{n}{{\left( {{{{(k+1)}}^{3}}-{{k}^{3}}} \right)}}={{(n+1)}^{3}}-1,\text{ maka}\\{{\left( {n+1} \right)}^{3}}-1=3\sum\limits_{{k=1}}^{n}{{{{k}^{2}}}}+3\frac{1}{2}n\left( {n+1} \right)+n\end{array}$

$\displaystyle \begin{array}{l}\text{Berarti}\\3\sum\limits_{{k=1}}^{n}{{{{k}^{2}}}}={{\left( {n+1} \right)}^{3}}-\frac{3}{2}n\left( {n+1} \right)-n-1\\=\frac{1}{2}\cdot 2{{(n+1)}^{3}}-\frac{1}{2}\cdot 3n(n+1)-2\cdot \frac{1}{2}(n+1)\\=\frac{1}{2}(n+1)\left( {2{{{(n+1)}}^{2}}-3n-2} \right)\\=\frac{1}{2}(n+1)\left( {2{{n}^{2}}+n} \right)\\=\frac{1}{2}(n+1)n(2n+1)\\\text{Jadi }\sum\limits_{{k=1}}^{n}{{{{k}^{2}}}}=\frac{1}{6}n(n+1)(2n+1)\end{array}$

[STMIK] Latihan UAS Kalkulus 2018

June 26, 2018June 26, 2018 cloudrain1 Uncategorized

https://drive.google.com/open?id=1KhDC_VuZfUU83QQ8_BEWxth-7W99dN6W

STMIK Latihan 2 UAS Desain Algo

January 9, 2018 cloudrain1 Uncategorized

Silakan dikerjakan.

https://drive.google.com/file/d/18ePbR1e1artFZI2rvyI1IzKs7ii1M2vF/view?usp=drivesdk

[STMIK] Latihan UAS Analisis dan Desain Algoritma – 2017/2018

December 19, 2017 cloudrain1 Uncategorized

Materi Brute Force dan Divide and Conquer

Silakan dikerjakan.

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[STMIK] Latihan UAS 2017

December 18, 2017December 18, 2017 cloudrain1 Uncategorized

Latihan UAS ini bisa di download di sini.
Continue reading →

[STMIK] [Pemrograman Dasar] Latihan UAS

December 21, 2016 cloudrain1 Uncategorized

download di sini
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Author: cloudrain1

Contoh soal Induksi

Pembuktian rumus penjumlahan bilangan kuadrat berurutan

[STMIK] Latihan UAS Kalkulus 2018

STMIK Latihan 2 UAS Desain Algo

[STMIK] Latihan UAS Analisis dan Desain Algoritma – 2017/2018

[STMIK] Latihan UAS 2017

[STMIK] [Pemrograman Dasar] Latihan UAS

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