Selingan (Barisan & Deret)

Lanjutan contoh soal dan pembahasannya terkait barisan dan deret

$\begin{array}{ll}\\ 1.&\textrm{Hasil kali bilangan bentuk berikut}\\ &\left ( 1-\displaystyle \frac{1}{4} \right )\left ( 1-\displaystyle \frac{1}{5} \right )\left ( 1-\displaystyle \frac{1}{6} \right )\cdots \left ( 1-\displaystyle \frac{1}{100} \right )\\ &\textrm{adalah}\: ....\\ &\textrm{A}.\quad \displaystyle \frac{1}{100} \: \: \qquad\qquad\qquad\qquad\qquad  \textrm{D}.\quad \displaystyle \frac{4}{100}\\\\ &\textrm{B}.\quad \displaystyle \frac{2}{100}\qquad\qquad \color{black}\textrm{C}.\quad \displaystyle \color{red}\frac{3}{100}\qquad\quad \color{black}\textrm{E}.\quad \displaystyle \frac{5}{100}\\\\ &\textbf{Jawab}:\\ &\textrm{Perhatikan bahwa}\\ &\begin{aligned}&\left ( 1-\displaystyle \frac{1}{4} \right )\left ( 1-\displaystyle \frac{1}{5} \right )\left ( 1-\displaystyle \frac{1}{6} \right )\cdots \left ( 1-\displaystyle \frac{1}{100} \right )\\ &=\left ( \displaystyle \frac{3}{4} \right )\left ( \displaystyle \frac{4}{5} \right )\left ( \displaystyle \frac{5}{6} \right )\cdots \left ( \displaystyle \frac{99}{100} \right )\\ &=\left ( \displaystyle \frac{\color{red}3}{\not{4}} \right )\left ( \displaystyle \frac{\not{4}}{\not{5}} \right )\left ( \displaystyle \frac{\not{5}}{\not{6}} \right )\cdots \left ( \displaystyle \frac{\not{99}}{\color{red}100} \right )\\ &=\color{red}\displaystyle \frac{3}{100} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 2.&\textrm{Hasil kali bilangan bentuk berikut}\\ &\left ( 1-\displaystyle \frac{2}{3} \right )\left ( 1-\displaystyle \frac{2}{5} \right )\left ( 1-\displaystyle \frac{2}{7} \right )\cdots \left ( 1-\displaystyle \frac{2}{2023} \right )\\ &\textrm{adalah}\: ....\\ &\textrm{A}.\quad \color{red}\displaystyle \frac{1}{2023} \: \: \qquad\qquad\qquad\qquad\qquad\: \:   \textrm{D}.\quad \displaystyle \frac{4}{2023}\\\\ &\textrm{B}.\quad \displaystyle \frac{2}{2023}\qquad\qquad \color{black}\textrm{C}.\quad \displaystyle \frac{3}{2023}\qquad\quad \color{black}\textrm{E}.\quad \displaystyle \frac{5}{2023}\\\\ &\textbf{Jawab}:\\ &\textrm{Dengan cara pembahasan pada no.1 di atas, maka}\\ &\begin{aligned}&\left ( 1-\displaystyle \frac{2}{3} \right )\left ( 1-\displaystyle \frac{2}{5} \right )\left ( 1-\displaystyle \frac{2}{7} \right )\cdots \left ( 1-\displaystyle \frac{2}{2023} \right )\\ &=\left ( \displaystyle \frac{1}{3} \right )\left ( \displaystyle \frac{3}{5} \right )\left ( \displaystyle \frac{5}{7} \right )\cdots \left ( \displaystyle \frac{2021}{2023} \right )\\ &=\left ( \displaystyle \frac{\color{red}1}{\not{3}} \right )\left ( \displaystyle \frac{\not{3}}{\not{5}} \right )\left ( \displaystyle \frac{\not{5}}{\not{7}} \right )\cdots \left ( \displaystyle \frac{\not{2021}}{\color{red}2023} \right )\\ &=\color{red}\displaystyle \frac{1}{2023} \end{aligned} \end{array}$.

 $\begin{array}{ll}\\ 3.&\textrm{Bentuk sederhana dari}\\ &\displaystyle \frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\cdots +\frac{1}{2022\times 2023}\\\\ &\textbf{Pembahasan}:\\ &\begin{aligned}&=\left ( 1-\displaystyle \frac{1}{2} \right )+\left ( \displaystyle \frac{1}{2}-\frac{1}{3} \right )+\left ( \displaystyle \frac{1}{3}-\frac{1}{4} \right )+\\ & \qquad\qquad +\quad\cdots\qquad +\left ( \displaystyle \frac{1}{2022}-\frac{1}{2023} \right )\\ &=1-\displaystyle \frac{1}{2023}\\ &=\color{blue}\displaystyle \frac{2022}{2023} \end{aligned}  \end{array}$.

$\begin{array}{|c|}\hline \begin{aligned}\color{blue}\textbf{Seb}&\color{blue}\textbf{agai pengingat kita}\\ \bullet \quad&1+2+3+\cdots +n=\displaystyle \frac{n(n+1)}{2}\\ \bullet \quad&1+3+5+\cdots +(2n-1)=n^{2}\\ \bullet \quad&1^{2}+2^{2}+3^{2}+\cdots +n^{2}=\displaystyle \frac{n(n+1)(2n+1)}{6}\\ \bullet \quad&1^{3}+2^{3}+3^{3}+\cdots +n^{3}=\left ( \displaystyle \frac{n(n+1)}{2} \right )^{2}\\ \bullet \quad&1+\displaystyle \frac{1}{1+2}+\frac{1}{1+2+3}+\cdots +\frac{1}{1+2+3+\cdots +n}=\displaystyle \frac{2n}{n+1}\\ \bullet \quad&\displaystyle \frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\cdots +\frac{1}{n(n+1)}=\displaystyle \frac{n}{n+1}\\ \bullet \quad&\displaystyle \frac{1}{1.2.3}+\frac{1}{2.3.4}+\cdots +\frac{1}{n(n+1)(n+2)}=\displaystyle \frac{n(n+3)}{4(n+1)(n+2)} \end{aligned}\\\hline \end{array}$.

$\begin{array}{ll}\\ 4.&\textrm{Bentuk sederhana dari}\\ &\displaystyle \frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+\cdots +\frac{1}{\sqrt{2023}-\sqrt{2022}}\\\\ &\textbf{Pembahasan}:\\ &\begin{aligned}\displaystyle \frac{1}{1+\sqrt{2}}&=\frac{1}{1+\sqrt{2}}\times \frac{1-\sqrt{2}}{1-\sqrt{2}}\\ &=\displaystyle \frac{1-\sqrt{2}}{1^{2}-(\sqrt{2})^{2}}=\frac{1-\sqrt{2}}{1-2}\\ &=\displaystyle \frac{1-\sqrt{2}}{-1}=\color{red}\sqrt{2}-1\\ \displaystyle \frac{1}{\sqrt{2}+\sqrt{3}}&=\frac{1}{\sqrt{2}+\sqrt{3}}\times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}\\ &=\displaystyle \frac{\sqrt{2}-\sqrt{3}}{(\sqrt{2})^{2}-(\sqrt{3})^{2}}=\frac{\sqrt{2}-\sqrt{3}}{2-3}\\ &=\displaystyle \frac{\sqrt{2}-\sqrt{3}}{-1}=\color{red}\sqrt{3}-\sqrt{2}\\ \displaystyle \frac{1}{\sqrt{3}+\sqrt{4}}&=\frac{1}{\sqrt{3}+\sqrt{4}}\times \frac{\sqrt{3}-\sqrt{4}}{\sqrt{3}-\sqrt{4}}\\ &=\displaystyle \frac{\sqrt{3}-\sqrt{4}}{(\sqrt{3})^{2}-(\sqrt{4})^{2}}=\frac{\sqrt{3}-\sqrt{4}}{3-4}\\ &=\displaystyle \frac{\sqrt{3}-\sqrt{4}}{-1}=\color{red}\sqrt{4}-\sqrt{3}\\ &\vdots \\  \end{aligned}\\ &\textrm{Sehingga}\\ &\begin{aligned}&=\color{red}\not{\sqrt{2}}-1\color{black}+\color{red}\not{\sqrt{3}}-\not{\sqrt{2}}\color{black}+\color{red}\not{\sqrt{4}}-\not{\sqrt{3}}\color{black}+\color{red}\cdots \color{black}+\color{red}\sqrt{2023}-\not{\sqrt{2022}}\\ &=\color{blue}\sqrt{2023}-1 \end{aligned}  \end{array}$