Contoh Soal 3 Induksi Matematika (Matematika Wajib Kelas XI)

 $\begin{array}{ll}\\ 11.&\textrm{Perhatikanlah pernyataan-pernyataan berikut}\\ &(1)\quad \displaystyle \sum_{i=1}^{5}(5i+2)=4\displaystyle \sum_{i=1}^{5}i+10\\ &(2)\quad \displaystyle \sum_{i=1}^{5}(5i^{2}-i)=5\displaystyle \sum_{i=1}^{5}i^{2}-\sum_{i=1}^{5}i\\ &(3)\quad \displaystyle \sum_{i=1}^{5}(3i-4)=3\displaystyle \sum_{i=1}^{5}i^{2}-4\\ &(4)\quad \displaystyle \sum_{i=1}^{5}(i+7i^{2})=\displaystyle \sum_{i=1}^{5}i-7\sum_{i=1}^{5}i\\ &\textrm{Pernyataan yang tepat ditunjukkan oleh}....\\ &\begin{array}{llll}\\ \color{red}\textrm{a}.&(1)\: \: \textrm{dan}\: \: (2)\\ \textrm{b}.&(1)\: \: \textrm{dan}\: \: (3)\\ \textrm{c}.&(1)\: \: \textrm{dan}\: \: (4)\\ \textrm{d}.&(2)\: \: \textrm{dan}\: \: (3)\\ \textrm{e}.&(2)\: \: \textrm{dan}\: \: (4) \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{a}\\ &\color{blue}\begin{aligned}(1)\quad&\displaystyle \sum_{i=1}^{5}(5i+2)=4\displaystyle \sum_{i=1}^{5}i+\sum_{i=1}^{5}2\\ &=4\displaystyle \sum_{i=1}^{5}i+5\times 2\\ &=4\displaystyle \sum_{i=1}^{5}i+10\\ (2)\quad&\displaystyle \sum_{i=1}^{5}(5i^{2}-i)=5\displaystyle \sum_{i=1}^{5}i^{2}-\sum_{i=1}^{5}i\\ (3)\quad&\displaystyle \sum_{i=1}^{5}(3i-4)=3\displaystyle \sum_{i=1}^{5}i^{2}-\sum_{i=1}^{5}4\\ &=3\displaystyle \sum_{i=1}^{5}i^{2}-5\times 4\\ &=3\displaystyle \sum_{i=1}^{5}i^{2}-20\\ (4)\quad&\displaystyle \sum_{i=1}^{5}(i+7i^{2})=\displaystyle \sum_{i=1}^{5}i+7\sum_{i=1}^{5}i \end{aligned} \end{array}$

$\begin{array}{ll}\\ 12.&\textrm{Hasil dari}\: \: \displaystyle \sum_{i=1}^{4}i^{2}+\sum_{i=5}^{6}i^{2}\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\displaystyle 86\\ \color{red}\textrm{b}.&\displaystyle 91\\ \textrm{c}.&\displaystyle 95\\ \textrm{d}.&\displaystyle 101\\ \textrm{e}.&\displaystyle 105 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{b}\\ &\color{blue}\begin{aligned}\displaystyle \sum_{i=1}^{4}i^{2}+\sum_{i=5}^{6}i^{2}&=\displaystyle \sum_{i=1}^{6}i^{2}\\ &=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}\\ &=1+4+9+16+25+36\\ &=91 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 13.&\textrm{Hasil dari}\: \: \displaystyle \sum_{i=2}^{5}\left ( 4i^{2}-2i \right )\: \: \textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\displaystyle 144\\ \textrm{b}.&\displaystyle 148\\ \textrm{c}.&\displaystyle 154\\ \textrm{d}.&\displaystyle 164\\ \color{red}\textrm{e}.&\displaystyle 188 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{e}\\ &\color{blue}\begin{aligned}&\displaystyle \sum_{i=2}^{5}\left ( 4i^{2}-2i \right )\\ &=\left ( 4.2^{2}-2.2 \right )+\left ( 4.3^{2}-2.3 \right )+\left ( 4.4^{2}-2.4 \right )+\left ( 4.5^{2}-2.5 \right )\\ &=12+30+56+90\\ &=188 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 14.&\textrm{Bentuk}\: \: 11^{n}-1\: \: \textrm{dengan}\: \: n\: \: \textrm{bilangan asli}\\ &\textrm{akan habis dibagi oleh}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\displaystyle 7\\ \textrm{b}.&\displaystyle 9\\ \color{red}\textrm{c}.&\displaystyle 10\\ \textrm{d}.&\displaystyle 11\\ \textrm{e}.&\displaystyle 13 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\color{blue}\begin{aligned}\textrm{Bentuk}&\: \: 11^{n}-1\\ \textrm{untuk}&\: \: n=1\\ &=11^{1}-1\\ &=10 \end{aligned} \end{array}$

$\begin{array}{ll}\\ 15.&\textrm{Rumus yang tepat untuk pola}\: \: 12,13,14,15,...\\ &\textrm{adalah}\: ....\\ &\begin{array}{llll}\\ \textrm{a}.&\displaystyle U_{n}=n+9\\ \textrm{b}.&\displaystyle U_{n}=n+10\\ \color{red}\textrm{c}.&\displaystyle U_{n}=n+11\\ \textrm{d}.&\displaystyle U_{n}=2n+10\\ \textrm{e}.&\displaystyle U_{n}=2n+11 \end{array}\\\\ &\textrm{Jawab}:\quad \color{red}\textbf{c}\\ &\color{blue}\begin{aligned}\textrm{Bentuk}&\: \: 12,13,14,15,...\\ \textrm{untuk}&\: \: U_{n}=pn+q\\ 12&=p+q\\ 13&=2p+q\\ \textrm{akan}&\: \textrm{didapatkan}\\ &\begin{cases} p & =1 \\ q & =11 \end{cases}\\ \textrm{Sehing}&\textrm{ga}\\ U_{n}&=n+11 \end{aligned} \end{array}$