Deret Tak Hingga : Soal dan Pembahasan

Berikut ini adalah kumpulan soal mengenai deret tak hingga. Jika ada jawaban yang salah, mohon dikoreksi melalui komentar. Terima kasih.

1. \dfrac13+\dfrac2{3^2}+\dfrac3{3^3}+\dfrac4{3^4}+\cdots= ....
   
   Misal
   $\begin{array}{rl}s& ={\displaystyle \frac{1}{3}}+{\displaystyle \frac{2}{3^2}}+{\displaystyle \frac{3}{3^3}}+{\displaystyle \frac{4}{3^4}}+\cdots \\ {\displaystyle \frac{1}{3}}s& ={\displaystyle \frac{1}{3^2}}+{\displaystyle \frac{2}{3^3}}+{\displaystyle \frac{3}{3^4}}+{\displaystyle \frac{4}{3^5}}+\cdots & -\\ {\displaystyle \frac{2}{3}}s& ={\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{3^2}}+{\displaystyle \frac{1}{3^3}}+{\displaystyle \frac{1}{3^4}}+\cdots \\ {\displaystyle \frac{2}{3}}s& ={\displaystyle \frac{{\displaystyle \frac{1}{3}}}{1-{\displaystyle \frac{1}{3}}}}\\ {\displaystyle \frac{2}{3}}s& ={\displaystyle \frac{{\displaystyle \frac{1}{3}}}{{\displaystyle \frac{2}{3}}}}\\ {\displaystyle \frac{2}{3}}s& ={\displaystyle \frac{1}{2}}\\ s& ={\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{3}{2}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle {\displaystyle \frac{3}{4}}}}}}\end{array}$

   Jadi, \dfrac13+\dfrac2{3^2}+\dfrac3{3^3}+\dfrac4{3^4}+\cdots=\dfrac34.
   
   
2. 8+\dfrac{16}3+\dfrac83+\dfrac{32}{27}+\dfrac{40}{81}+\cdots=a\times\dfrac67. Maka nilai a adalah....
   
   1. 21
   2. 20

   3. 19
   4. 18
   Misal
   

   $\begin{array}{rl}s& =8+{\displaystyle \frac{16}{3}}+{\displaystyle \frac{8}{3}}+{\displaystyle \frac{32}{27}}+{\displaystyle \frac{40}{81}}+\cdots \\ s& =8+{\displaystyle \frac{16}{3}}+{\displaystyle \frac{24}{9}}+{\displaystyle \frac{32}{27}}+{\displaystyle \frac{40}{81}}+\cdots \\ s& =8+{\displaystyle \frac{16}{3}}+{\displaystyle \frac{24}{3^2}}+{\displaystyle \frac{32}{3^3}}+{\displaystyle \frac{40}{3^4}}+\cdots \\ {\displaystyle \frac{1}{3}}s& ={\displaystyle \frac{8}{3}}+{\displaystyle \frac{16}{3^2}}+{\displaystyle \frac{24}{3^3}}+{\displaystyle \frac{32}{3^4}}+{\displaystyle \frac{40}{3^5}}+\cdots & -\\ {\displaystyle \frac{2}{3}}s& =8+{\displaystyle \frac{8}{3}}+{\displaystyle \frac{8}{3^2}}+{\displaystyle \frac{8}{3^3}}+{\displaystyle \frac{8}{3^4}}+\cdots \\ & ={\displaystyle \frac{8}{1-{\displaystyle \frac{1}{3}}}}\\ & ={\displaystyle \frac{8}{{\displaystyle \frac{2}{3}}}}\\ & =12\\ s& =12\cdot {\displaystyle \frac{3}{2}}\\ a\times {\displaystyle \frac{6}{7}}& =18\\ a& =18\cdot {\displaystyle \frac{7}{6}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 21}}}}\end{array}$

   Jadi, nilai a adalah 21.
   JAWAB: A
   
   
3. 
   

j2