Science:Math Exam Resources/Courses/MATH101/April 2016/Question 01 (d)/Solution 1

Note: A Geometric Series has the form $a+ar+ar^{2}+ar^{3}+...,$ where $a$ is the first term, and $r$ is called the common ratio. We can compute this series using the formula

$a+ar+ar^{2}+ar^{3}+...={\frac {\color {OrangeRed}a}{1-{\color {OliveGreen}r}}}$

We first write $0.321321321...$ in the Geometric Series form. Indeed,

${\begin{aligned}0.321321321...&=0.321+0.000321+0.000000321+...\\&={\frac {321}{10^{3}}}+{\frac {321}{10^{6}}}+{\frac {321}{10^{9}}}+....\end{aligned}}$

We can see that the first term is ${\color {OrangeRed}a={\frac {321}{10^{3}}}}$ and the common ration is ${\color {OliveGreen}r}={\frac {\text{the Second term}}{\text{the First term}}}={\frac {\frac {321}{10^{6}}}{\frac {321}{10^{3}}}}{\color {OliveGreen}={\frac {1}{10^{3}}}.}$.

Therefore,

${\begin{aligned}0.321321321...&={\frac {321}{10^{3}}}+{\frac {321}{10^{6}}}+{\frac {321}{10^{9}}}+....\\&={\frac {\color {OrangeRed}{\frac {321}{10^{3}}}}{1-{\color {OliveGreen}{\frac {1}{10^{3}}}}}}={\frac {\frac {321}{10^{3}}}{\frac {999}{10^{3}}}}\\&={\frac {321}{999}}={\color {blue}{\frac {107}{333}}}\end{aligned}}$

So, the answer is Z.