Contoh Soal 3 Induksi Matematika (Matematika Wajib Kelas XI)

 $\begin{array}{l}\\ 11.& \text{Perhatikanlah pernyataan-pernyataan berikut}\\ & (1){\displaystyle \sum _{i=1}^5(5i+2)=4{\displaystyle \sum _{i=1}^{5}i+10}}\\ & (2){\displaystyle \sum _{i=1}^5(5i^2-i)=5{\displaystyle \sum _{i=1}^5{i}^2-\sum _{i=1}^{5}i}}\\ & (3){\displaystyle \sum _{i=1}^5(3i-4)=3{\displaystyle \sum _{i=1}^5{i}^2-4}}\\ & (4){\displaystyle \sum _{i=1}^5(i+7i^2)={\displaystyle \sum _{i=1}^{5}i-7\sum _{i=1}^{5}i}}\\ & \text{Pernyataan yang tepat ditunjukkan oleh}....\\ & \begin{array}{l}\\ \text{a}.& (1)\text{dan}(2)\\ \text{b}.& (1)\text{dan}(3)\\ \text{c}.& (1)\text{dan}(4)\\ \text{d}.& (2)\text{dan}(3)\\ \text{e}.& (2)\text{dan}(4)\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{a}}\\ & \begin{array}{rl}(1)& {\displaystyle \sum _{i=1}^5(5i+2)=4{\displaystyle \sum _{i=1}^{5}i+\sum _{i=1}^{5}2}}\\ & =4{\displaystyle \sum _{i=1}^{5}i+5\times 2}\\ & =4{\displaystyle \sum _{i=1}^{5}i+10}\\ (2)& {\displaystyle \sum _{i=1}^5(5i^2-i)=5{\displaystyle \sum _{i=1}^5{i}^2-\sum _{i=1}^{5}i}}\\ (3)& {\displaystyle \sum _{i=1}^5(3i-4)=3{\displaystyle \sum _{i=1}^5{i}^2-\sum _{i=1}^{5}4}}\\ & =3{\displaystyle \sum _{i=1}^5{i}^2-5\times 4}\\ & =3{\displaystyle \sum _{i=1}^5{i}^2-20}\\ (4)& {\displaystyle \sum _{i=1}^5(i+7i^2)={\displaystyle \sum _{i=1}^{5}i+7\sum _{i=1}^{5}i}}\end{array}\end{array}$

$\begin{array}{l}\\ 12.& \text{Hasil dari}{\displaystyle \sum _{i=1}^4{i}^2+\sum _{i=5}^6{i}^2}\\ & \text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle 86}\\ \text{b}.& {\displaystyle 91}\\ \text{c}.& {\displaystyle 95}\\ \text{d}.& {\displaystyle 101}\\ \text{e}.& {\displaystyle 105}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{b}}\\ & \begin{array}{rl}{\displaystyle \sum _{i=1}^4{i}^2+\sum _{i=5}^6{i}^2}& ={\displaystyle \sum _{i=1}^6{i}^2}\\ & =1^2+2^2+3^2+4^2+5^2+6^2\\ & =1+4+9+16+25+36\\ & =91\end{array}\end{array}$

$\begin{array}{l}\\ 13.& \text{Hasil dari}{\displaystyle \sum _{i=2}^5(4i^2-2i)\text{adalah}....}\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle 144}\\ \text{b}.& {\displaystyle 148}\\ \text{c}.& {\displaystyle 154}\\ \text{d}.& {\displaystyle 164}\\ \text{e}.& {\displaystyle 188}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}& {\displaystyle \sum _{i=2}^5(4i^2-2i)}\\ & =({4.2}^2-2.2)+({4.3}^2-2.3)+({4.4}^2-2.4)+({4.5}^2-2.5)\\ & =12+30+56+90\\ & =188\end{array}\end{array}$

$\begin{array}{l}\\ 14.& \text{Bentuk}{11}^n-1\text{dengan}n\text{bilangan asli}\\ & \text{akan habis dibagi oleh}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle 7}\\ \text{b}.& {\displaystyle 9}\\ \text{c}.& {\displaystyle 10}\\ \text{d}.& {\displaystyle 11}\\ \text{e}.& {\displaystyle 13}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}\text{Bentuk}& {11}^n-1\\ \text{untuk}& n=1\\ & ={11}^1-1\\ & =10\end{array}\end{array}$

$\begin{array}{l}\\ 15.& \text{Rumus yang tepat untuk pola}12,13,14,15,...\\ & \text{adalah}....\\ & \begin{array}{l}\\ \text{a}.& {\displaystyle U_n=n+9}\\ \text{b}.& {\displaystyle U_n=n+10}\\ \text{c}.& {\displaystyle U_n=n+11}\\ \text{d}.& {\displaystyle U_n=2n+10}\\ \text{e}.& {\displaystyle U_n=2n+11}\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}\text{Bentuk}& 12,13,14,15,...\\ \text{untuk}& U_n=pn+q\\ 12& =p+q\\ 13& =2p+q\\ \text{akan}& \text{didapatkan}\\ & \{\begin{array}{ll}p& =1\\ q& =11\end{array}\\ \text{Sehing}& \text{ga}\\ U_n& =n+11\end{array}\end{array}$