Contoh Soal 1 Induksi Matematika (Matematika Wajib Kelas XI)

$\begin{array}{l}\\ 1.& \text{Hasil dari}{\displaystyle \sum _{i=1}^{6}16i\text{adalah}....}\\ & \begin{array}{l}\\ \text{a}.& 306\\ \text{b}.& 314\\ \text{c}.& 326\\ \text{d}.& 336\\ \text{e}.& 402\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}{\displaystyle \sum _{i=1}^{6}16i}& =16.1+16.2+16.3+16.4+16.5+16.6\\ & =16+32+48+64+80+96\\ & =336\end{array}\end{array}$

$\begin{array}{l}\\ 2.& \text{Hasil dari}{\displaystyle \sum _{i=2}^9{i}^2\text{adalah}....}\\ & \begin{array}{l}\\ \text{a}.& 274\\ \text{b}.& 278\\ \text{c}.& 280\\ \text{d}.& 284\\ \text{e}.& 286\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{d}}\\ & \begin{array}{rl}{\displaystyle \sum _{i=2}^9{i}^2}& =2^2+3^2+4^2+5^2+..+9^2\\ & =4+9+16+25+...+81\\ & =284\end{array}\end{array}$

$\begin{array}{l}\\ 3.& \text{Poa bilangan}12,14,16,18,20,...,(2n+10).\\ & \text{Nilai suku ke-100 adalah}....\\ & \begin{array}{l}\\ \text{a}.& 180\\ \text{b}.& 194\\ \text{c}.& 198\\ \text{d}.& 208\\ \text{e}.& 210\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{e}}\\ & \begin{array}{rl}{U}_n& =2n+10\\ U_{100}& =2\times 100+10\\ & =210\end{array}\end{array}$

$\begin{array}{l}\\ 4.& \text{Diketahui bahwa jika}\\ & 31+39+47+\cdots +8n+23=4n^2+27n\\ & \text{dengan}k,n\in \mathbb{N}\text{maka}\\ & 31+39+47+\cdots +8n+23+8k+31=....\\ & \begin{array}{l}\\ \text{a}.& 4k^2+27k\\ \text{b}.& 4k^2+35k\\ \text{c}.& 4k^2+35k+31\\ \text{d}.& 4k^2+35k+1\\ \text{e}.& 4k^2+35k+54\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}& \underset{4k^2+27k}{\underbrace{31+39+47+\cdots +8k+23}}+8k+31\\ & =4k^2+27k+8k+31\\ & =4k^2+35k+31\end{array}\end{array}$

$\begin{array}{l}\\ 5.& \text{Dengan Induksi Matematika untuk}n\in \mathbb{N}\\ & \text{dapat dibuktikan bahwa}n(n+1)(n+2)\\ & \text{akan habis dibagi oleh}\\ & \begin{array}{l}\\ \text{a}.& 4\\ \text{b}.& 5\\ \text{c}.& 6\\ \text{d}.& 7\\ \text{e}.& 8\end{array}\\ \\ & \text{Jawab}:\mathbf{\text{c}}\\ & \begin{array}{rl}P(n)& =n(n+1)(n+2)\\ P(1)& =1(1+1)(1+2)=1.2.3\\ & \text{adalah bilangan yang habis dibagi 6}\end{array}\end{array}$